At the dawn of quantum theory, Heisenberg, Schrödinger, Bohr and Pauli were embroiled in a dispute over whether trajectories of particles, defined by their positions over time, could exist. The argument against trajectories was based on an apparent paradox: To draw a “line” depicting a trajectory of a particle along a path implies that there is a momentum vector that carries the particle along that path. But a line is a one-dimensional curve through space, and since at any point in time the particle’s position is perfectly localized, then by Heisenberg’s uncertainty principle, it can have no definable momentum to carry it along.

My previous blog shows the way out of this paradox, by assembling wavepackets that are spread in both space and momentum, explicitly obeying the uncertainty principle. This is nothing new to anyone who has taken a quantum course. But the surprising thing is that in some potentials, like a harmonic potential, the wavepacket travels without broadening, just like classical particles on a trajectory. A dramatic demonstration of this can be seen in this YouTube video. But other potentials “break up” the wavepacket, especially potentials that display classical chaos. Because phase space is one of the best tools for studying classical chaos, especially Hamiltonian chaos, it can be enlisted to dig deeper into the question of the quantum trajectory—not just about the existence of a quantum trajectory, but why quantum systems retain a shadow of their classical counterparts.

Phase Space

Phase space is the state space of Hamiltonian systems. Concepts of phase space were first developed by Boltzmann as he worked on the problem of statistical mechanics. Phase space was later codified by Gibbs for statistical mechanics and by Poincare for orbital mechanics, and it was finally given its name by Paul and Tatiana Ehrenfest (a husband-wife team) in correspondence with the German physicist Paul Hertz (See Chapter 6, “The Tangled Tale of Phase Space”, in Galileo Unbound by D. D. Nolte (Oxford, 2018)).

The stretched-out phase-space functions … are very similar to the stochastic layer that forms in separatrix chaos in classical systems.

The idea of phase space is very simple for classical systems: it is just a plot of the momentum of a particle as a function of its position. For a given initial condition, the trajectory of a particle through its natural configuration space (for instance our 3D world) is traced out as a path through phase space. Because there is one momentum variable per degree of freedom, then the dimensionality of phase space for a particle in 3D is 6D, which is difficult to visualize. But for a one-dimensional dynamical system, like a simple harmonic oscillator (SHO) oscillating in a line, the phase space is just two-dimensional, which is easy to see. The phase-space trajectories of an SHO are simply ellipses, and if the momentum axis is scaled appropriately, the trajectories are circles. The particle trajectory in phase space can be animated just like a trajectory through configuration space as the position and momentum change in time p(x(t)). For the SHO, the point follows the path of a circle going clockwise.

*Fig. 1 Phase space of the simple harmonic oscillator. The “orbits” have constant energy.*

A more interesting phase space is for the simple pendulum, shown in Fig. 2. There are two types of orbits: open and closed. The closed orbits near the origin are like those of a SHO. The open orbits are when the pendulum is spinning around. The dividing line between the open and closed orbits is called a separatrix. Where the separatrix intersects itself is a saddle point. This saddle point is the most important part of the phase space portrait: it is where chaos emerges when perturbations are added.

Fig. 2 Phase space for a simple pendulum. For small amplitudes the orbits are closed like those of a SHO. For large amplitudes the orbits become open as the pendulum spins about its axis. (Reproduced from Introduction to Modern Dynamics, 2nd Ed., pg. )

One route to classical chaos is through what is known as “separatrix chaos”. It is easy to see why saddle points (also known as hyperbolic points) are the source of chaos: as the system trajectory approaches the saddle, it has two options of which directions to go. Any additional degree of freedom in the system (like a harmonic drive) can make the system go one way on one approach, and the other way on another approach, mixing up the trajectories. An example of the stochastic layer of separatrix chaos is shown in Fig. 3 for a damped driven pendulum. The chaotic behavior that originates at the saddle point extends out along the entire separatrix.

*Fig. 3 The stochastic layer of separatrix chaos for a damped driven pendulum. (Reproduced from Introduction to Modern Dynamics, 2nd Ed., pg. )*

The main question about whether or not there is a quantum trajectory depends on how quantum packets behave as they approach a saddle point in phase space. Since packets are spread out, it would be reasonable to assume that parts of the packet will go one way, and parts of the packet will go another. But first, one has to ask: Is a phase-space description of quantum systems even possible?

Quantum Phase Space: The Wigner Distribution Function

Phase-space portraits are arguably the most powerful tool in the toolbox of classical dynamics, and one would like to retain its uses for quantum systems. However, there is that pesky paradox about quantum trajectories that cannot admit the existence of one-dimensional curves through such a phase space. Furthermore, there is no direct way of taking a wavefunction and simply “finding” its position or momentum to plot points on such a quantum phase space.

The answer was found in 1932 by Eugene Wigner (1902 – 1905), an Hungarian physicist working at Princeton. He realized that it was impossible to construct a quantum probability distribution in phase space that had positive values everywhere. This is a problem, because negative probabilities have no direct interpretation. But Wigner showed that if one relaxed the requirements a bit, so that expectation values computed over some distribution function (that had positive and negative values) gave correct answers that matched experiments, then this distribution function would “stand in” for an actual probability distribution.

The distribution function that Wigner found is called the Wigner distribution function. Given a wavefunction ψ(x), the Wigner distribution is defined as

*Fig. 4 Wigner distribution function in (x, p) phase space.*

The Wigner distribution function is the Fourier transform of the convolution of the wavefunction. The pure position dependence of the wavefunction is converted into a spread-out position-momentum function in phase space. For a Gaussian wavefunction ψ(x) with a finite width in space, the W-function in phase space is a two-dimensional Gaussian with finite widths in both space and momentum. In fact, the Δx-Δp product of the W-function is precisely the uncertainty production of the Heisenberg uncertainty relation.

The question of the quantum trajectory from the phase-space perspective becomes whether a Wigner function behaves like a localized “packet” that evolves in phase space in a way analogous to a classical particle, and whether classical chaos is reflected in the behavior of quantum systems.

The Harmonic Oscillator

The quantum harmonic oscillator is a rare and special case among quantum potentials, because the energy spacings between all successive states are all the same. This makes it possible for a Gaussian wavefunction, which is a superposition of the eigenstates of the harmonic oscillator, to propagate through the potential without broadening. To see an example of this, watch the first example in this YouTube video for a Schrödinger cat state in a two-dimensional harmonic potential. For this very special potential, the Wigner distribution behaves just like a (broadened) particle on an orbit in phase space, executing nice circular orbits.

A comparison of the classical phase-space portrait versus the quantum phase-space portrait is shown in Fig. 5. Where the classical particle is a point on an orbit, the quantum particle is spread out, obeying the Δx-Δp Heisenberg product, but following the same orbit as the classical particle.

Fig. 5 Classical versus quantum phase-space portraits for a harmonic oscillator. For a classical particle, the trajectory is a point executing an orbit. For a quantum particle, the trajectory is a Wigner distribution that follows the same orbit as the classical particle.

However, a significant new feature appears in the Wigner representation in phase space when there is a coherent superposition of two states, known as a “cat” state, after Schrödinger’s cat. This new feature has no classical analog. It is the coherent interference pattern that appears at the zero-point of the harmonic oscillator for the Schrödinger cat state. There is no such thing as “classical” coherence, so this feature is absent in classical phase space portraits.

Two examples of Wigner distributions are shown in Fig. 6 for a statistical (incoherent) mixture of packets and a coherent superposition of packets. The quantum coherence signature is present in the coherent case but not the statistical mixture case. The coherence in the Wigner distribution represents “off-diagonal” terms in the density matrix that leads to interference effects in quantum systems. Quantum computing algorithms depend critically on such coherences that tend to decay rapidly in real-world physical systems, known as decoherence, and it is possible to make statements about decoherence by watching the zero-point interference.

Fig. 6 Quantum phase-space portraits of double wave packets. On the left, the wave packets have no coherence, being a statistical mixture. On the right is the case for a coherent superposition, or “cat state” for two wave packets in a one-dimensional harmonic oscillator.

Whereas Gaussian wave packets in the quantum harmonic potential behave nearly like classical systems, and their phase-space portraits are almost identical to the classical phase-space view (except for the quantum coherence), most quantum potentials cause wave packets to disperse. And when saddle points are present in the classical case, then we are back to the question about how quantum packets behave as they approach a saddle point in phase space.

Quantum Pendulum and Separatrix Chaos

One of the simplest anharmonic oscillators is the simple pendulum. In the classical case, the period diverges if the pendulum gets very close to going vertical. A similar thing happens in the quantum case, but because the motion has strong anharmonicity, an initial wave packet tends to spread dramatically as parts of the wavefunction less vertical stretch away from the part of the wave function that is more nearly vertical. Fig. 7 is a snap-shot about a eighth of a period after the wave packet was launched. The packet has already stretched out along the separatrix. A double-cat-state was used, so there is a second packet that has coherent interference with the first. To see a movie of the time evolution of the wave packet and the orbit in quantum phase space, see the YouTube video.

Fig. 7 Wavefunction of a quantum pendulum released near vertical. The phase-space portrait is very similar to the classical case, except that the phase-space distribution is stretched out along the separatrix. The initial state for the phase-space portrait was a cat state.

The simple pendulum does have a saddle point, but it is degenerate because the angle is modulo -2-pi. A simple potential that has a non-degenerate saddle point is a double-well potential.

Quantum Double-Well and Separatrix Chaos

The symmetric double-well potential has a saddle point at the mid-point between the two well minima. A wave packet approaching the saddle will split into to packets that will follow the individual separatrixes that emerge from the saddle point (the unstable manifolds). This effect is seen most dramatically in the middle pane of Fig. 8. For the full video of the quantum phase-space evolution, see this YouTube video. The stretched-out distribution in phase space is highly analogous to the separatrix chaos seen for the classical system.

Fig. 8 Phase-space portraits of the Wigner distribution for a wavepacket in a double-well potential. The packet approaches the central saddle point, where the probability density splits along the unstable manifolds.

Conclusion

A common statement often made about quantum chaos is that quantum systems tend to suppress chaos, only exhibiting chaos for special types of orbits that produce quantum scars. However, from the phase-space perspective, the opposite may be true. The stretched-out Wigner distribution functions, for critical wave packets that interact with a saddle point, are very similar to the stochastic layer that forms in separatrix chaos in classical systems. In this sense, the phase-space description brings out the similarity between classical chaos and quantum chaos.

By David D. Nolte Sept. 25, 2022

YouTube Video

YouTube Video of Dynamics in Quantum Phase Space

References

1. T. Curtright, D. Fairlie, C. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space. (World Scientific, New Jersey, 2014).

2. J. R. Nagel, A Review and Application of the Finite-Difference Time-Domain Algorithm Applied to the Schrödinger Equation, ACES Journal, Vol. 24, NO. 1, pp. 1-8 (2009)

September 4, 2022March 19, 2023 by David D. Nolte

Is There a Quantum Trajectory?

Heisenberg’s uncertainty principle is a law of physics – it cannot be violated under any circumstances, no matter how much we may want it to yield or how hard we try to bend it. Heisenberg, as he developed his ideas after his lone epiphany like a monk on the isolated island of Helgoland off the north coast of Germany in 1925, became a bit of a zealot, like a religious convert, convinced that all we can say about reality is a measurement outcome. In his view, there was no independent existence of an electron other than what emerged from a measuring apparatus. Reality, to Heisenberg, was just a list of numbers in a spread sheet—matrix elements. He took this line of reasoning so far that he stated without exception that there could be no such thing as a trajectory in a quantum system. When the great battle commenced between Heisenberg’s matrix mechanics against Schrödinger’s wave mechanics, Heisenberg was relentless, denying any reality to Schrödinger’s wavefunction other than as a calculation tool. He was so strident that even Bohr, who was on Heisenberg’s side in the argument, advised Heisenberg to relent [1]. Eventually a compromise was struck, as Heisenberg’s uncertainty principle allowed Schrödinger’s wave functions to exist within limits—his uncertainty limits.

Disaster in the Poconos

Yet the idea of an actual trajectory of a quantum particle remained a type of heresy within the close quantum circles. Years later in 1948, when a young Richard Feynman took the stage at a conference in the Poconos, he almost sabotaged his career in front of Bohr and Dirac—two of the giants who had invented quantum mechanics—by having the audacity to talk about particle trajectories in spacetime diagrams.

Feynman was making his first presentation of a new approach to quantum mechanics that he had developed based on path integrals. The challenge was that his method relied on space-time graphs in which “unphysical” things were allowed to occur. In fact, unphysical things were required to occur, as part of the sum over many histories of his path integrals. For instance, a key element in the approach was allowing electrons to travel backwards in time as positrons, or a process in which the electron and positron annihilate into a single photon, and then the photon decays back into an electron-positron pair—a process that is not allowed by mass and energy conservation. But this is a possible history that must be added to Feynman’s sum.

It all looked like nonsense to the audience, and the talk quickly derailed. Dirac pestered him with questions that he tried to deflect, but Dirac persisted like a raven. A question was raised about the Pauli exclusion principle, about whether an orbital could have three electrons instead of the required two, and Feynman said that it could—all histories were possible and had to be summed over—an answer that dismayed the audience. Finally, as Feynman was drawing another of his space-time graphs showing electrons as lines, Bohr rose to his feet and asked derisively whether Feynman had forgotten Heisenberg’s uncertainty principle that made it impossible to even talk about an electron trajectory.

It was hopeless. The audience gave up and so did Feynman as the talk just fizzled out. It was a disaster. What had been meant to be Feynman’s crowning achievement and his entry to the highest levels of theoretical physics, had been a terrible embarrassment. He slunk home to Cornell where he sank into one of his depressions. At the close of the Pocono conference, Oppenheimer, the reigning king of physics, former head of the successful Manhattan Project and newly selected to head the prestigious Institute for Advanced Study at Princeton, had been thoroughly disappointed by Feynman.

But what Bohr and Dirac and Oppenheimer had failed to understand was that as long as the duration of unphysical processes was shorter than the energy differences involved, then it was literally obeying Heisenberg’s uncertainty principle. Furthermore, Feynman’s trajectories—what became his famous “Feynman Diagrams”—were meant to be merely cartoons—a shorthand way to keep track of lots of different contributions to a scattering process. The quantum processes certainly took place in space and time, conceptually like a trajectory, but only so far as time durations, and energy differences and locations and momentum changes were all within the bounds of the uncertainty principle. Feynman had invented a bold new tool for quantum field theory, able to supply deep results quickly. But no one at the Poconos could see it.

Coherent States

When Feynman had failed so miserably at the Pocono conference, he had taken the stage after Julian Schwinger, who had dazzled everyone with his perfectly scripted presentation of quantum field theory—the competing theory to Feynman’s. Schwinger emerged the clear winner of the contest. At that time, Roy Glauber (1925 – 2018) was a young physicist just taking his PhD from Schwinger at Harvard, and he later received a post-doc position at Princeton’s Institute for Advanced Study where he became part of a miniature revolution in quantum field theory that revolved around—not Schwinger’s difficult mathematics—but Feynman’s diagrammatic method. So Feynman won in the end. Glauber then went on to Caltech, where he filled in for Feynman’s lectures when Feynman was off in Brazil playing the bongos. Glauber eventually returned to Harvard where he was already thinking about the quantum aspects of photons in 1956 when news of the photon correlations in the Hanbury-Brown Twiss (HBT) experiment were published. Three years later, when the laser was invented, he began developing a theory of photon correlations in laser light that he suspected would be fundamentally different than in natural chaotic light.

Because of his background in quantum field theory, and especially quantum electrodynamics, it was fairly easy to couch the quantum optical properties of coherent light in terms of Dirac’s creation and annihilation operators of the electromagnetic field. Glauber developed a “coherent state” operator that was a minimum uncertainty state of the quantized electromagnetic field, related to the minimum-uncertainty wave functions derived initially by Schrödinger in the late 1920’s. The coherent state represents a laser operating well above the lasing threshold and behaved as “the most classical” wavepacket that can be constructed. Glauber was awarded the Nobel Prize in Physics in 2005 for his work on such “Glauber states” in quantum optics.

Quantum Trajectories

Glauber’s coherent states are built up from the natural modes of a harmonic oscillator. Therefore, it should come as no surprise that these coherent-state wavefunctions in a harmonic potential behave just like classical particles with well-defined trajectories. The quadratic potential matches the quadratic argument of the the Gaussian wavepacket, and the pulses propagate within the potential without broadening, as in Fig. 3, showing a snapshot of two wavepackets propagating in a two-dimensional harmonic potential. This is a somewhat radical situation, because most wavepackets in most potentials (or even in free space) broaden as they propagate. The quadratic potential is a special case that is generally not representative of how quantum systems behave.

Fig. 3 Harmonic potential in 2D and two examples of pairs of pulses propagating without broadening. The wavepackets in the center are oscillating in line, and the wavepackets on the right are orbiting the center of the potential in opposite directions. (Movies of the quantum trajectories can be viewed at Physics Unbound.)

To illustrate this special status for the quadratic potential, the wavepackets can be launched in a potential with a quartic perturbation. The quartic potential is anharmonic—the frequency of oscillation depends on the amplitude of oscillation unlike for the harmonic oscillator, where amplitude and frequency are independent. The quartic potential is integrable, like the harmonic oscillator, and there is no avenue for chaos in the classical analog. Nonetheless, wavepackets broaden as they propagate in the quartic potential, eventually spread out into a ring in the configuration space, as in Fig. 4.

Fig. 4 Potential with a quartic corrections. The initial gaussian pulses spread into a “ring” orbiting the center of the potential.

A potential with integrability has as many conserved quantities to the motion as there are degrees of freedom. Because the quartic potential is integrable, the quantum wavefunction may spread, but it remains highly regular, as in the “ring” that eventually forms over time. However, integrable potentials are the exception rather than the rule. Most potentials lead to nonintegrable motion that opens the door to chaos.

A classic (and classical) potential that exhibits chaos in a two-dimensional configuration space is the famous Henon-Heiles potential. This has a four-dimensional phase space which admits classical chaos. The potential has a three-fold symmetry which is one reason it is non-integral, since a particle must “decide” which way to go when it approaches a saddle point. In the quantum regime, wavepackets face the same decision, leading to a breakup of the wavepacket on top of a general broadening. This allows the wavefunction eventually to distribute across the entire configuration space, as in Fig. 5.

Fig. 5 The Henon-Heiles two-dimensional potential supports Hamiltonian chaos in the classical regime. In the quantum regime, the wavefunction spreads to eventually fill the accessible configuration space (for constant energy).

Youtube Video

Movies of quantum trajectories can be viewed at my Youtube Channel, Physics Unbound. The answer to the question “Is there a quantum trajectory?” can be seen visually as the movies run—they do exist in a very clear sense under special conditions, especially coherent states in a harmonic oscillator. And the concept of a quantum trajectory also carries over from a classical trajectory in cases when the classical motion is integrable, even in cases when the wavefunction spreads over time. However, for classical systems that display chaotic motion, wavefunctions that begin as coherent states break up into chaotic wavefunctions that fill the accessible configuration space for a given energy. The character of quantum evolution of coherent states—the most classical of quantum wavefunctions—in these cases reflects the underlying character of chaotic motion in the classical analogs. This process can be seen directly watching the movies as a wavepacket approaches a saddle point in the potential and is split. Successive splits of the multiple wavepackets as they interact with the saddle points is what eventually distributes the full wavefunction into its chaotic form.

Therefore, the idea of a “quantum trajectory”, so thoroughly dismissed by Heisenberg, remains a phenomenological guide that can help give insight into the behavior of quantum systems—both integrable and chaotic.

As a side note, the laws of quantum physics obey time-reversal symmetry just as the classical equations do. In the third movie of “A Quantum Ballet“, wavefunctions in a double-well potential are tracked in time as they start from coherent states that break up into chaotic wavefunctions. It is like watching entropy in action as an ordered state devolves into a disordered state. But at the half-way point of the movie, the imaginary part of the wavefunction has its sign flipped, and the dynamics continue. But now the wavefunctions move from disorder into an ordered state, seemingly going against the second law of thermodynamics. Flipping the sign of the imaginary part of the wavefunction at just one instant in time plays the role of a time-reversal operation, and there is no violation of the second law.

By David D. Nolte, Sept. 4, 2022

YouTube Video

YouTube Video of Quantum Trajectories

References

[1] See Chapter 8 , On the Quantum Footpath, in Galileo Unbound, D. D. Nolte (Oxford University Press, 2018)

[2] J. R. Nagel, A Review and Application of the Finite-Difference Time-Domain Algorithm Applied to the Schrödinger Equation, ACES Journal, Vol. 24, NO. 1, pp. 1-8 (2009)

August 14, 2022March 19, 2023 by David D. Nolte

Quantum Chaos and the Cheshire Cat

Alice’s disturbing adventures in Wonderland tumbled upon her like a string of accidents as she wandered a world of chaos. Rules were never what they seemed and shifted whenever they wanted. She even met a cat who grinned ear-to-ear and could disappear entirely, or almost entirely, leaving only its grin.

The vanishing Cheshire Cat reminds us of another famous cat—Arnold’s Cat—that introduced the ideas of stretching and folding of phase-space volumes in non-integrable Hamiltonian systems. But when Arnold’s Cat becomes a Quantum Cat, a central question remains: What happens to the chaotic behavior of the classical system … does it survive the transition to quantum mechanics? The answer is surprisingly like the grin of the Cheshire Cat—the cat vanishes, but the grin remains. In the quantum world of the Cheshire Cat, the grin of the classical cat remains even after the rest of the cat vanished.

The Cheshire Cat fades away, leaving only its grin, like a fine filament, as classical chaos fades into quantum, leaving behind a quantum scar.

The Quantum Mechanics of Classically Chaotic Systems

The simplest Hamiltonian systems are integrable—they have as many constants of the motion as degrees of freedom. This holds for quantum systems as well as for classical. There is also a strong correspondence between classical and quantum systems for the integrable cases—literally the Correspondence Principle—that states that quantum systems at high quantum number approach classical behavior. Even at low quantum numbers, classical resonances are mirrored by quantum eigenfrequencies that can show highly regular spectra.

But integrable systems are rare—surprisingly rare. Almost no real-world Hamiltonian system is integrable, because the real world warps the ideal. No spring can displace indefinitely, and no potential is perfectly quadratic. There are always real-world non-idealities that destroy one constant of the motion or another, opening the door to chaos.

When classical Hamiltonian systems become chaotic, they don’t do it suddenly. Almost all transitions to chaos in Hamiltonian systems are gradual. One of the best examples of this is the KAM theory that starts with invariant action integrals that generate invariant tori in phase space. As nonintegrable perturbations increase, the tori break up slowly into island chains of stability as chaos infiltrates the separatrixes—first as thin filaments of chaos surrounding the islands—then growing in width to take up more and more of phase space. Even when chaos is fully developed, small islands of stability can remain—the remnants of stable orbits of the unperturbed system.

When the classical becomes quantum, chaos softens. Quantum wave functions don’t like to be confined—they spread and they tunnel. The separatrix of classical chaos—that barrier between regions of phase space—cannot constrain the exponential tails of wave functions. And the origin of chaos itself—the homoclinic point of the separatrix—gets washed out. Then the regular orbits of the classical system reassert themselves, and they appear, like the vestige of the Cheshire Cat, as a grin.

The Quantum Circus

The empty stadium is a surprisingly rich dynamical system that has unexpected structure in both the classical and the quantum domain. Its importance in classical dynamics comes from the fact that its periodic orbits are unstable and its non-periodic orbits are ergodic (filling all available space if given long enough). The stadium itself is empty so that particles (classical or quantum) are free to propagate between reflections from the perfectly-reflecting walls of the stadium. The ergodicity comes from the fact that the stadium—like a classic Roman chariot-race stadium, also known as a circus—is not a circle, but has a straight stretch between two half circles. This simple modification takes the stable orbits of the circle into the unstable orbits of the stadium.

A single classical orbit in a stadium is shown in Fig 1. This is an ergodic orbit that is non-periodic and eventually would fill the entire stadium space. There are other orbits that are nearly periodic, such as one that bounces back and forth vertically between the linear portions, but even this orbit will eventually wander into the circular part of the stadium and then become ergodic. The big quantum-classical question is what happens to these classical orbits when the stadium is shrunk to the nanoscale?

Fig. 1 A classical trajectory in a stadium. It will eventually visit every point, a property known as ergodicity.

Simulating an evolving quantum wavefunction in free space is surprisingly simple. Given a beginning quantum wavefunction A(x,y,t₀), the discrete update equation is

Perfect reflection from the boundaries of the stadium are incorporated through imposing a boundary condition that sends the wavefunction to zero. Simple!

A snap-shot of a wavefunction evolving in the stadium is shown in Fig. 2. To see a movie of the time evolution, see my YouTube episode.

Fig. 2 Snapshot of a quantum wavefunction in the stadium. (From YouTube)

The time average of the wavefunction after a long time has passed is shown in Fig. 3. Other than the horizontal nodal line down the center of the stadium, there is little discernible structure or symmetry. This is also true for the mean squared wavefunction shown in Fig. 4, although there is some structure that may be emerging in the semi-circular regions.

Fig. 3 Time-average wavefunction after a long time.

Fig. 4 Time-average of the squared wavefunction after a long time.

On the other hand, for special initial conditions that have a lot of symmetry, something remarkable happens. Fig. 5 shows several mean-squared results for special initial conditions. There is definite structure in these cases that were given the somewhat ugly name “quantum scars” in the 1980’s by Eric Heller who was one of the first to study this phenomenon [1].

Fig. 5 Quantum scars reflect periodic (but unstable) orbits of the classical system. Quantum effects tend to quench chaos and favor regular motion.

One can superpose highly-symmetric classical trajectories onto the figures, as shown in the bottom row. All of these classical orbits go through a high-symmetry point, such as the center of the stadium (on the left image) and through the focal point of the circular mirrors (in the other two images). The astonishing conclusion of this exercise is that the highly-symmetric periodic classical orbits remain behind as quantum scars—like the Cheshire Cat’s grin—when the system is in the quantum realm. The classical orbits that produce quantum scars have the important property of being periodic but unstable. A slight perturbation from the symmetric trajectory causes it to eventually become ergodic (chaotic). These scars are regions with enhanced probability density, what might be termed “quantum trajectories”, but do not show strong interference patterns.

It is important to make the distinction that it is also possible to construct special wavefunctions that are strictly periodic, such as a wave bouncing perfectly vertically between the straight portions. This leads to large-scale interference patterns that are not the same as the quantum scars.

Quantum Chaos versus Laser Speckle

In addition to the bouncing-wave cases that do not strictly produce quantum scars, there is another “neutral” phenomenon that produces interference patterns that look a lot like scars, but are simply the random addition of lots of plane waves with the same wavelength [2]. A snapshot in time of one of these superpositions is shown in Fig. 6. To see how the waves add together, see the YouTube channel episode.

Fig. 6 The sum of 100 randomly oriented plane waves of constant wavelength. (A snapshot from YouTube.)

By David D. Nolte, Aug. 14, 2022

YouTube Video

YouTube Video of Quantum Chaos

References

[1] Heller E J, Bound-state eigenfunctions of classically chaotic hamiltonian-systems – scars of periodic-orbits, Physical Review Letters 53 ,1515 (1984)

[2] Gutzwiller M C, Chaos in classical and quantum mechanics (New York: New York : Springer-Verlag, 1990)

February 28, 2022February 9, 2023 by David D. Nolte

Life in a Solar System with a Super-sized Jupiter

There are many known super-Jupiters that orbit their stars—they are detected through a slight Doppler wobble they induce on their stars [1]. But what would become of a rocky planet also orbiting those stars as they feel the tug of both the star and the super planet?

This is not of immediate concern for us, because our solar system has had its current configuration of planets for over 4 billion years. But there can be wandering interstellar planets or brown dwarfs that could visit our solar system, like Oumuamua did in 2017, but much bigger and able to scramble the planetary orbits. Such hypothesized astronomical objects have been given the name “Nemesis“, and it warrants thought on what living in an altered solar system might be like.

What would happen to Earth if Jupiter were 50 times bigger? Could we survive?

The Three-Body Problem

The Sun-Earth-Jupiter configuration is a three-body problem that has a long and interesting history, playing a key role in several aspects of modern dynamics [2]. There is no general analytical solution to the three-body problem. To find the behavior of three mutually interacting bodies requires numerical solution. However, there are subsets of the three-body problem that do yield to partial analytical approaches. One of these is called the restricted three-body problem [3]. It consists of two massive bodies plus a third (nearly) massless body that all move in a plane. This restricted problem was first tackled by Euler and later by Poincaré, who discovered the existence of chaos in its solutions.

The geometry of the restricted three-body problem is shown in Fig. 1. In this problem, take mass m₁ = m_S to be the Sun’s mass, m₂ = m_J to be Jupiter’s mass, and the third (small) mass is the Earth.

Fig. 1 The restricted 3-body problem in the plane. The third mass is negligible relative to the first two masses that obey 2-body dynamics.

The equation of motion for the Earth is

where

and the parameter ξ characterizes the strength of the perturbation of the Earth’s orbit around the Sun. The parameters for the Jupiter-Sun system are

with

for the 11.86 year journey of Jupiter around the Sun. Eq. (1) is a four-dimensional non-autonomous flow

The solutions of an Earth orbit are shown in Fig.2. The natural Earth-Sun-Jupiter system has a mass ratio m_J/m_S = 0.001 for Jupiter relative to the Sun mass. Even in this case, Jupiter causes perturbations of the Earth’s orbit by about one percent. If the mass of Jupiter increases, the perturbations would grow larger until around ξ= 0.06 when the perturbations become severe and the orbit grows unstable. The Earth gains energy from the momentum of the Sun-Jupiter system and can reach escape velocity. The simulation for a mass ratio of 0.07 shows the Earth ejected from the Solar System.

Fig.2 Orbit of Earth as a function of the size of a Jupiter-like planet. The natural system has a Jupiter-Earth mass ratio of 0.03. As the size of Jupiter increases, the Earth orbit becomes unstable and can acquire escape velocity to escape from the Solar System. From body3.m. (Reprinted from Ref. [4])

The chances for ejection depends on initial conditions for these simulations, but generally the danger becomes severe when Jupiter is about 50 times larger than it currently is. Otherwise the Earth remains safe from ejection. However, if the Earth is to keep its climate intact, then Jupiter should not be any larger than about 5 times its current size. At the other extreme, for a planet 70 times larger than Jupiter, the Earth may not get ejected at once, but it can take a wild ride through the solar system. A simulation for a 70x Jupiter is shown in Fig. 3. In this case, the Earth is captured for a while as a “moon” of Jupiter in a very tight orbit around the super planet as it orbits the sun before it is set free again to orbit the sun in highly elliptical orbits. Because of the premise of the restricted three-body problem, the Earth has no effect on the orbit of Jupiter.

Fig. 3 Orbit of Earth for T_J = 11.86 years and ξ = 0.069. The radius of Jupiter is R_J = 5.2. Earth is “captured” for a while by Jupiter into a very tight orbit.

Resonance

If Nemesis were to swing by and scramble the solar system, then Jupiter might move closer to the Earth. More ominously, the period of Jupiter’s orbit could come into resonance with the Earth’s period. This occurs when the ratio of orbital periods is a ratio of small integers. Resonance can amplify small perturbations, so perhaps Jupiter would become a danger to Earth. However, the forces exerted by Jupiter on the Earth changes the Earth’s orbit and hence its period, preventing strict resonance to occur, and the Earth is not ejected from the solar system even for initial rational periods or larger planet mass. This is related to the famous KAM theory of resonances by Kolmogorov, Arnold and Moser that tends to protect the Earth from the chaos of the solar system. More often than not in these scenarios, the Earth is either captured by the super Jupiter, or it is thrown into a large orbit that is still bound to the sun. Some examples are given in the following figures.

Fig. 5 Earth orbit for T_J = 12 years and ξ = 0.071. The Earth is thrown into a nearly circular orbit beyond the orbit of Saturn.

Fig. 6 Earth Orbit for T_J = 4 years and ξ = 0.0615. Earth is thrown into an orbit of high ellipticity out to the orbit of Neptune.

Life on a planet in a solar system with two large bodies has been envisioned in dramatic detail in the science fiction novel “Three-Body Problem” by Liu Cixin about the Trisolarians of the closest known exoplanet to Earth–Proxima Centauri b.

By David D. Nolte, Feb. 28, 2022

Matlab Code: body3.m

function body3

clear

chsi0 = 1/1000;     % Earth-moon ratio = 1/317
wj0 = 2*pi/11.86;

wj = 2*pi/8;
chsi = 73*chsi0;    % (11.86,60) (11.86,67.5) (11.86,69) (11.86,70) (4,60) (4,61.5) (8,73) (12,71) 

rj = 5.203*(wj0/wj)^0.6666

rsun = chsi*rj/(1+chsi);
rjup = (1/chsi)*rj/(1+1/chsi);

r0 = 1-rsun;
y0 = [r0 0 0 2*pi/sqrt(r0)];

tspan = [0 300];
options = odeset('RelTol',1e-5,'AbsTol',1e-6);
[t,y] = ode45(@f5,tspan,y0,options);

figure(1)
plot(t,y(:,1),t,y(:,3))

figure(2)
plot(y(:,1),y(:,3),'k')
axis equal
axis([-6 6 -6 6])

RE = sqrt(y(:,1).^2 + y(:,3).^2);
stdRE = std(RE)

%print -dtiff -r800 threebody

    function yd = f5(t,y)
        
        xj = rjup*cos(wj*t);
        yj = rjup*sin(wj*t);
        xs = -rsun*cos(wj*t);
        ys = -rsun*sin(wj*t);
        rj32 = ((y(1) - xj).^2 + (y(3) - yj).^2).^1.5;
        r32 = ((y(1) - xs).^2 + (y(3) - ys).^2).^1.5;

        yp(1) = y(2);
        yp(2) = -4*pi^2*((y(1)-xs)/r32 + chsi*(y(1)-xj)/rj32);
        yp(3) = y(4);
        yp(4) = -4*pi^2*((y(3)-ys)/r32 + chsi*(y(3)-yj)/rj32);
 
        yd = [yp(1);yp(2);yp(3);yp(4)];

    end     % end f5

end

References:

[1] D. D. Nolte, “The Fall and Rise of the Doppler Effect,” Physics Today, vol. 73, no. 3, pp. 31-35, Mar (2020)

[2] J. Barrow-Green, Poincaré and the three body problem. London Mathematical Society, 1997.

[3] M. C. Gutzwiller, “Moon-Earth-Sun: The oldest three-body problem,” Reviews of Modern Physics, vol. 70, no. 2, pp. 589-639, Apr (1998)

[4] D. D. Nolte, Introduction to Modern Dynamics : Chaos, Networks, Space and Time, 1st ed. (Oxford University Press, 2015).

September 22, 2021December 17, 2021 by David D. Nolte

Spontaneous Symmetry Breaking: A Mechanical Model

Symmetry is the canvas upon which the laws of physics are written. Symmetry defines the invariants of dynamical systems. But when symmetry breaks, the laws of physics break with it, sometimes in dramatic fashion. Take the Big Bang, for example, when a highly-symmetric form of the vacuum, known as the “false vacuum”, suddenly relaxed to a lower symmetry, creating an inflationary cascade of energy that burst forth as our Universe.

The early universe was extremely hot and energetic, so much so that all the forces of nature acted as one–described by a unified Lagrangian (as yet resisting discovery by theoretical physicists) of the highest symmetry. Yet as the universe expanded and cooled, the symmetry of the Lagrangian broke, and the unified forces split into two (gravity and electro-nuclear). As the universe cooled further, the Lagrangian (of the Standard Model) lost more symmetry as the electro-nuclear split into the strong nuclear force and the electro-weak force. Finally, at a tiny fraction of a second after the Big Bang, the universe cooled enough that the unified electro-week force broke into the electromagnetic force and the weak nuclear force. At each stage, spontaneous symmetry breaking occurred, and invariants of physics were broken, splitting into new behavior. In 2008, Yoichiro Nambu received the Nobel Prize in physics for his model of spontaneous symmetry breaking in subatomic physics.

Fig. 1 The spontanous symmetry breaking cascade after the Big Bang. From Ref.

Bifurcation Physics

Physics is filled with examples of spontaneous symmetry breaking. Crystallization and phase transitions are common examples. When the temperature is lowered on a fluid of molecules with high average local symmetry, the molecular interactions can suddenly impose lower-symmetry constraints on relative positions, and the liquid crystallizes into an ordered crystal. Even solid crystals can undergo a phase transition as one symmetry becomes energetically advantageous over another, and the crystal can change to a new symmetry.

In mechanics, any time a potential function evolves slowly with some parameter, it can start with one symmetry and evolve to another lower symmetry. The mechanical system governed by such a potential may undergo a discontinuous change in behavior.

In complex systems and chaos theory, sudden changes in behavior can be quite common as some parameter is changed continuously. These discontinuous changes in behavior, in response to a continuous change in a control parameter, is known as a bifurcation. There are many types of bifurcation, carrying descriptive names like the pitchfork bifurcation, period-doubling bifurcation, Hopf bifurcation, and fold bifurcation, among others. The pitchfork bifurcation is a typical example, shown in Fig. 2. As a parameter is changed continuously (horizontal axis), a stable fixed point suddenly becomes unstable and two new stable fixed points emerge at the same time. This type of bifurcation is called pitchfork because the diagram looks like a three-tined pitchfork. (This is technically called a supercritical pitchfork bifurcation. In a subcritical pitchfork bifurcation the solid and dashed lines are swapped.) This is exactly the bifurcation displayed by a simple mechanical model that illustrates spontaneous symmetry breaking.

Fig. 2 Bifurcation plot of a pitchfork bifurcation. As a parameter is changed smoothly and continuously (horizontal axis), a stable fixed point suddenly splits into three fixed points: one unstable and the other two stable.

Sliding Mass on a Rotating Hoop

One of the simplest mechanical models that displays spontaneous symmetry breaking and the pitchfork bifurcation is a bead sliding without friction on a circular hoop that is spinning on the vertical axis, as in Fig. 3. When it spins very slowly, this is just a simple pendulum with a stable equilibrium at the bottom, and it oscillates with a natural oscillation frequency ω₀ = sqrt(g/b), where b is the radius of the hoop and g is the acceleration due to gravity. On the other hand, when it spins very fast, then the bead is flung to to one side or the other by centrifugal force. The bead then oscillates around one of the two new stable fixed points, but the fixed point at the bottom of the hoop is very unstable, because any deviation to one side or the other will cause the centrifugal force to kick in. (Note that in the body frame, centrifugal force is a non-inertial force that arises in the non-inertial coordinate frame. )

Fig. 3 A bead sliding without friction on a circular hoop rotating about a vertical axis. At high speed, the bead has a stable equilibrium to either side of the vertical.

The solution uses the Euler equations for the body frame along principal axes. In order to use the standard definitions of ω₁, ω₂, and ω₃, the angle θ MUST be rotated around the x-axis. This means the x-axis points out of the page in the diagram. The y-axis is tilted up from horizontal by θ, and the z-axis is tilted from vertical by θ. This establishes the body frame.

The components of the angular velocity are

And the moments of inertia are (assuming the bead is small)

There is only one Euler equation that is non-trivial. This is for the x-axis and the angle θ. The x-axis Euler equation is

and solving for the angular acceleration gives.

This is a harmonic oscillator with a “phase transition” that occurs as ω increases from zero. At first the stable equilibrium is at the bottom. But when ω passes a critical threshold, the equilibrium angle begins to increase to a finite angle set by the rotation speed.

This can only be real if the magnitude of the argument is equal to or less than unity, which sets the critical threshold spin rate to make the system move to the new stable points to one side or the other for

which interestingly is the natural frequency of the non-rotating pendulum. Note that there are two equivalent angles (positive and negative), so this problem has a degeneracy.

This is an example of a dynamical phase transition that leads to spontaneous symmetry breaking and a pitchfork bifurcation. By integrating the angular acceleration we can get the effective potential for the problem. One contribution to the potential is due to gravity. The other is centrifugal force. When combined and plotted in Fig. 4 for a family of values of the spin rate ω, a pitchfork emerges naturally by tracing the minima in the effective potential. The values of the new equilibrium angles are given in Fig. 2.

Fig. 4 Effective potential as a function of angle for a family of spin rates. At the transition spin rate, the effective potential is essentially flat with zero natural frequency. The pitchfork is the dashed green line.

Below the transition threshold for ω, the bottom of the hoop is the equilibrium position. To find the natural frequency of oscillation, expand the acceleration expression

For small oscillations the natural frequency is given by

As the effective potential gets flatter, the natural oscillation frequency decreases until it vanishes at the transition spin frequency. As the hoop spins even faster, the new equilibrium positions emerge. To find the natural frequency of the new equilibria, expand θ around the new equilibrium θ’ = θ – θ₀

Which is a harmonic oscillator with oscillation angular frequency

Note that this is zero frequency at the transition threshold, then rises to match the spin rate of the hoop at high frequency. The natural oscillation frequency as a function of the spin looks like Fig. 5.

Fig. 5 Angular oscillation frequency for the bead. The bifurcation occurs at the critical spin rate ω = sqrt(g/b).

This mechanical analog is highly relevant for the spontaneous symmetry breaking that occurs in ferroelectric crystals when they go through a ferroelectric transition. At high temperature, these crystals have no internal polarization. But as the crystal cools towards the ferroelectric transition temperature, the optical-mode phonon modes “soften” as the phonon frequency decreases and vanishes at the transition temperature when the crystal spontaneously polarizes in one of several equivalent directions. The observation of mode softening in a polar crystal is one signature of an impending ferroelectric phase transition. Our mass on the hoop captures this qualitative physics nicely.

Golden Behavior

For fun, let’s find at what spin frequency the harmonic oscillation frequency at the dynamic equilibria equal the original natural frequency of the pendulum. Then

which is the golden ratio. It’s spooky how often the golden ratio appears in random physics problems!

March 14, 2021March 16, 2021 by David D. Nolte

The Butterfly Effect versus the Divergence Meter: The Physics of Stein’s Gate

Imagine if you just discovered how to text through time, i.e. time-texting, when a close friend meets a shocking death. Wouldn’t you text yourself in the past to try to prevent it? But what if, every time you change the time-line and alter the future in untold ways, the friend continues to die, and you seemingly can never stop it? This is the premise of Stein’s Gate, a Japanese sci-fi animé bringing in the paradoxes of time travel, casting CERN as an evil clandestine spy agency, and introducing do-it-yourself inventors, hackers, and wacky characters, while it centers on a terrible death of a lovable character that can never be avoided.

It is also a good computational physics project that explores the dynamics of bifurcations, bistability and chaos. I teach a course in modern dynamics in the Physics Department at Purdue University. The topics of the course range broadly from classical mechanics to chaos theory, social networks, synchronization, nonlinear dynamics, economic dynamics, population dynamics, evolutionary dynamics, neural networks, special and general relativity, among others that are covered in the course using a textbook that takes a modern view of dynamics [1].

For the final project of the second semester the students (Junior physics majors) are asked to combine two or three of the topics into a single project. Students have come up with a lot of creative combinations: population dynamics of zombies, nonlinear dynamics of negative gravitational mass, percolation of misinformation in presidential elections, evolutionary dynamics of neural architecture, and many more. In that spirit, and for a little fun, in this blog I explore the so-called physics of Stein’s Gate.

Stein’s Gate and the Divergence Meter

Stein’s Gate is a Japanese TV animé series that had a world-wide distribution in 2011. The central premise of the plot is that certain events always occur even if you are on different timelines—like trying to avoid someone’s death in an accident.

This is the problem confronting Rintaro Okabe who tries to stop an accident that kills his friend Mayuri Shiina. But every time he tries to change time, she dies in some other way. It turns out that all the nearby timelines involve her death. According to a device known as The Divergence Meter, Rintaro must get farther than 4% away from the original timeline to have a chance to avoid the otherwise unavoidable event.

This is new. Usually, time-travel Sci-Fi is based on the Butterfly Effect. Chaos theory is characterized by something called sensitivity to initial conditions (SIC), meaning that slightly different starting points produce trajectories that diverge exponentially from nearby trajectories. It is called the Butterfly Effect because of the whimsical notion that a butterfly flapping its wings in China can cause a hurricane in Florida. In the context of the butterfly effect, if you go back in time and change anything at all, the effect cascades through time until the present time in unrecognizable. As an example, in one episode of the TV cartoon The Simpsons, Homer goes back in time to the age of the dinosaurs and kills a single mosquito. When he gets back to our time, everything has changed in bazaar and funny ways.

Stein’s Gate introduces a creative counter example to the Butterfly Effect. Instead of scrambling the future when you fiddle with the past, you find that you always get the same event, even when you change a lot of the conditions—Mayuri still dies. This sounds eerily familiar to a physicist who knows something about chaos theory. It means that the unavoidable event is acting like a stable fixed point in the time dynamics—an attractor! Even if you change the initial conditions, the dynamics draw you back to the fixed point—in this case Mayuri’s accident. What would this look like in a dynamical system?

The Local Basin of Attraction

Dynamical systems can be described as trajectories in a high-dimensional state space. Within state space there are special points where the dynamics are static—known as fixed points. For a stable fixed point, a slight perturbation away will relax back to the fixed point. For an unstable fixed point, on the other hand, a slight perturbation grows and the system dynamics evolve away. However, there can be regions in state space where every initial condition leads to trajectories that stay within that region. This is known as a basin of attraction, and the boundaries of these basins are called separatrixes.

A high-dimensional state space can have many basins of attraction. All the physics that starts within a basin stays within that basin—almost like its own self-consistent universe, bordered by countless other universes. There are well-known physical systems that have many basins of attraction. String theory is suspected to generate many adjacent universes where the physical laws are a little different in each basin of attraction. Spin glasses, which are amorphous solid-state magnets, have this property, as do recurrent neural networks like the Hopfield network. Basins of attraction occur naturally within the physics of these systems.

It is possible to embed basins of attraction within an existing dynamical system. As an example, let’s start with one of the simplest types of dynamics, a hyperbolic fixed point

that has a single saddle fixed point at the origin. We want to add a basin of attraction at the origin with a domain range given by a radius r₀. At the same time, we want to create a separatrix that keeps the outer hyperbolic dynamics separate from the internal basin dynamics. To keep all outer trajectories in the outer domain, we can build a dynamical barrier to prevent the trajectories from crossing the separatrix. This can be accomplished by adding a radial repulsive term

In x-y coordinates this is

We also want to keep the internal dynamics of our basin separate from the external dynamics. To do this, we can multiply by a sigmoid function, like a Heaviside function H(r-r₀), to zero-out the external dynamics inside our basin. The final external dynamics is then

Now we have to add the internal dynamics for the basin of attraction. To make it a little more interesting, let’s make the internal dynamics an autonomous oscillator

Putting this all together, gives

This looks a little complex, for such a simple model, but it illustrates the principle. The sigmoid is best if it is differentiable, so instead of a Heaviside function it can be a Fermi function

The phase-space portrait of the final dynamics looks like

Figure 1. Hyperbolic dynamics with a basin of attraction embedded inside it at the origin. The dynamics inside the basin of attraction is a limit cycle.

Adding the internal dynamics does not change the far-field external dynamics, which are still hyperbolic. The repulsive term does split the central saddle point into two saddle points, one on each side left-and-right, so the repulsive term actually splits the dynamics. But the internal dynamics are self-contained and separate from the external dynamics. The origin is an unstable spiral that evolves to a limit cycle. The basin boundary has marginal stability and is known as a “wall”.

To verify the stability of the external fixed point, find the fixed point coordinates

and evaluate the Jacobian matrix (for A = 1 and x₀ = 2)

which is clearly a saddle point because the determinant is negative.

In the context of Stein’s Gate, the basin boundary is equivalent to the 4% divergence which is necessary to escape the internal basin of attraction where Mayuri meets her fate.

Python Program: SteinsGate2D.py

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
SteinsGate2D.py
Created on Sat March 6, 2021

@author: David Nolte
Introduction to Modern Dynamics, 2nd edition (Oxford University Press, 2019)

2D simulation of Stein's Gate Divergence Meter
"""
import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt

plt.close('all')

def solve_flow(param,lim = [-6,6,-6,6],max_time=20.0):

    def flow_deriv(x_y, t0, alpha, beta, gamma):
        #"""Compute the time-derivative ."""
        x, y = x_y
        
        w = 1
        R2 = x**2 + y**2
        R = np.sqrt(R2)
        arg = (R-2)/0.1
        env1 = 1/(1+np.exp(arg))
        env2 = 1 - env1
        
        f = env2*(x*(1/(R-1.99)**2 + 1e-2) - x) + env1*(w*y + w*x*(1 - R))
        g = env2*(y*(1/(R-1.99)**2 + 1e-2) + y) + env1*(-w*x + w*y*(1 - R))
        
        return [f,g]
    model_title = 'Steins Gate'

    plt.figure()
    xmin = lim[0]
    xmax = lim[1]
    ymin = lim[2]
    ymax = lim[3]
    plt.axis([xmin, xmax, ymin, ymax])

    N = 24*4 + 47
    x0 = np.zeros(shape=(N,2))
    ind = -1
    for i in range(0,24):
        ind = ind + 1
        x0[ind,0] = xmin + (xmax-xmin)*i/23
        x0[ind,1] = ymin
        ind = ind + 1
        x0[ind,0] = xmin + (xmax-xmin)*i/23
        x0[ind,1] = ymax
        ind = ind + 1
        x0[ind,0] = xmin
        x0[ind,1] = ymin + (ymax-ymin)*i/23
        ind = ind + 1
        x0[ind,0] = xmax
        x0[ind,1] = ymin + (ymax-ymin)*i/23
    ind = ind + 1
    x0[ind,0] = 0.05
    x0[ind,1] = 0.05
    
    for thetloop in range(0,10):
        ind = ind + 1
        theta = 2*np.pi*(thetloop)/10
        ys = 0.125*np.sin(theta)
        xs = 0.125*np.cos(theta)
        x0[ind,0] = xs
        x0[ind,1] = ys

    for thetloop in range(0,10):
        ind = ind + 1
        theta = 2*np.pi*(thetloop)/10
        ys = 1.7*np.sin(theta)
        xs = 1.7*np.cos(theta)
        x0[ind,0] = xs
        x0[ind,1] = ys

    for thetloop in range(0,20):
        ind = ind + 1
        theta = 2*np.pi*(thetloop)/20
        ys = 2*np.sin(theta)
        xs = 2*np.cos(theta)
        x0[ind,0] = xs
        x0[ind,1] = ys
        
    ind = ind + 1
    x0[ind,0] = -3
    x0[ind,1] = 0.05
    ind = ind + 1
    x0[ind,0] = -3
    x0[ind,1] = -0.05
    ind = ind + 1
    x0[ind,0] = 3
    x0[ind,1] = 0.05
    ind = ind + 1
    x0[ind,0] = 3
    x0[ind,1] = -0.05
    ind = ind + 1
    x0[ind,0] = -6
    x0[ind,1] = 0.00
    ind = ind + 1
    x0[ind,0] = 6
    x0[ind,1] = 0.00
           
    colors = plt.cm.prism(np.linspace(0, 1, N))
                        
    # Solve for the trajectories
    t = np.linspace(0, max_time, int(250*max_time))
    x_t = np.asarray([integrate.odeint(flow_deriv, x0i, t, param)
                      for x0i in x0])

    for i in range(N):
        x, y = x_t[i,:,:].T
        lines = plt.plot(x, y, '-', c=colors[i])
        plt.setp(lines, linewidth=1)
       
    plt.show()
    plt.title(model_title)
        
    return t, x_t

param = (0.02,0.5,0.2)        # Steins Gate
lim = (-6,6,-6,6)

t, x_t = solve_flow(param,lim)

plt.savefig('Steins Gate')

The Lorenz Butterfly

Two-dimensional phase space cannot support chaos, and we would like to reconnect the central theme of Stein’s Gate, the Divergence Meter, with the Butterfly Effect. Therefore, let’s actually incorporate our basin of attraction inside the classic Lorenz Butterfly. The goal is to put an attracting domain into the midst of the three-dimensional state space of the Lorenz butterfly in a way that repels the butterfly, without destroying it, but attracts local trajectories. The question is whether the butterfly can survive if part of its state space is made unavailable to it.

The classic Lorenz dynamical system is

As in the 2D case, we will put in a repelling barrier that prevents external trajectories from moving into the local basin, and we will isolate the external dynamics by using the sigmoid function. The final flow equations looks like

where the radius is relative to the center of the attracting basin

and r₀ is the radius of the basin. The center of the basin is at [x₀, y₀, z₀] and we are assuming that x₀ = 0 and y₀ = 0 and z₀ = 25 for the standard Butterfly parameters p = 10, r = 25 and b = 8/3. This puts our basin of attraction a little on the high side of the center of the Butterfly. If we embed it too far inside the Butterfly it does actually destroy the Butterfly dynamics.

When r₀ = 0, the dynamics of the Lorenz’ Butterfly are essentially unchanged. However, when r₀ = 1.5, then there is a repulsive effect on trajectories that pass close to the basin. It can be seen as part of the trajectory skips around the outside of the basin in Figure 2.

Figure 2. The Lorenz Butterfly with part of the trajectory avoiding the basin that is located a bit above the center of the Butterfly.

Trajectories can begin very close to the basin, but still on the outside of the separatrix, as in the top row of Figure 3 where the basin of attraction with r₀ = 1.5 lies a bit above the center of the Butterfly. The Butterfly still exists for the external dynamics. However, any trajectory that starts within the basin of attraction remains there and executes a stable limit cycle. This is the world where Mayuri dies inside the 4% divergence. But if the initial condition can exceed 4%, then the Butterfly effect takes over. The bottom row of Figure 2 shows that the Butterfly itself is fragile. When the external dynamics are perturbed more strongly by more closely centering the local basin, the hyperbolic dynamics of the Butterfly are impeded and the external dynamics are converted to a stable limit cycle. It is interesting that the Butterfly, so often used as an illustration of sensitivity to initial conditions (SIC), is itself sensitive to perturbations that can convert it away from chaos and back to regular motion.

Figure 3. (Top row) A basin of attraction is embedded a little above the Butterfly. The Butterfly still exists for external trajectories, but any trajectory that starts inside the basin of attraction remains inside the basin. (Bottom row) The basin of attraction is closer to the center of the Butterfly and disrupts the hyperbolic point and converts the Butterfly into a stable limit cycle.

Discussion and Extensions

In the examples shown here, the local basin of attraction was put in “by hand” as an isolated region inside the dynamics. It would be interesting to consider more natural systems, like a spin glass or a Hopfield network, where the basins of attraction occur naturally from the physical principles of the system. Then we could use the “Divergence Meter” to explore these physical systems to see how far the dynamics can diverge before crossing a separatrix. These systems are impossible to visualize because they are intrinsically very high dimensional systems, but Monte Carlo approaches could be used to probe the “sizes” of the basins.

Another interesting extension would be to embed these complex dynamics into spacetime. Since this all started with the idea of texting through time, it would be interesting (and challenging) to see how we could describe this process in a high dimensional Minkowski space that had many space dimensions (but still only one time dimension). Certainly it would violate the speed of light criterion, but we could then take the approach of David Deutsch and view the time axis as if it had multiple branches, like the branches of the arctangent function, creating time-consistent sheets within a sheave of flat Minkowski spaces.

References

[1] D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd edition (Oxford University Press, 2019)

[2] E. N. Lorenz, The essence of chaos. (University of Washington Press, 1993)

[3] E. N. Lorenz, “Deterministic Nonperiodic Flow,” Journal of the Atmospheric Sciences, vol. 20, no. 2, pp. 130-141, 1963 (1963)

November 16, 2020 by David D. Nolte

Edward Lorenz’ Chaotic Butterfly

The butterfly effect is one of the most widely known principles of chaos theory. It has become a meme, propagating through popular culture in movies, books, TV shows and even casual conversation.

Can a butterfly flapping its wings in Florida send a hurricane to New York?

The origin of the butterfly effect is — not surprisingly — the image of a butterfly-like set of trajectories that was generated, in one of the first computer simulations of chaos theory, by Edward Lorenz.

Lorenz’ Butterfly

Excerpted from Galileo Unbound (Oxford, 2018) pg. 215

When Edward Lorenz (1917 – 2008) was a child, he memorized all perfect squares up to ten thousand. This obvious interest in mathematics led him to a master’s degree in the subject at Harvard in 1940 under the supervision of Georg Birkhoff. Lorenz’s master’s thesis was on an aspect of Riemannian geometry, but his foray into nonlinear dynamics was triggered by the intervention of World War II. Only a few months before receiving his doctorate in mathematics from Harvard, the Japanese bombed Pearl Harbor.

Lorenz left the PhD program at Harvard to join the United States Army Air Force to train as a weather forecaster in early 1942, and he took courses on forecasting and meteorology at MIT. After receiving a second master’s degree, this time in meteorology, Lorenz was posted to Hawaii, then to Saipan and finally to Guam. His area of expertise was in high-level winds, which were important for high-altitude bombing missions during the final months of the war in the Pacific. After the Japanese surrender, Lorenz returned to MIT, where he continued his studies in meteorology, receiving his doctorate degree in 1948 with a thesis on the application of fluid dynamical equations to predict the motion of storms.

One of Lorenz’ colleagues at MIT was Norbert Wiener (1894 – 1964), with whom he sometimes played chess during lunch at the faculty club. Wiener had published his landmark book Cybernetics: Control and Communication in the Animal and Machine in 1949 which arose out of the apparently mundane problem of gunnery control during the Second World War. As an abstract mathematician, Wiener attempted to apply his cybernetic theory to the complexities of weather, but he developed a theorem concerning nonlinear fluid dynamics which appeared to show that linear interpolation, of sufficient resolution, would suffice for weather forecasting, possibly even long-range forecasting. Many on the meteorology faculty embraced this theorem because it fell in line with common practices of the day in which tomorrow’s weather was predicted using linear regression on measurements taken today. However, Lorenz was skeptical, having acquired a detailed understanding of atmospheric energy cascades as larger vortices induced smaller vortices all the way down to the molecular level, dissipating as heat, and then all the way back up again as heat drove large-scale convection. This was clearly not a system that would yield to linearization. Therefore, Lorenz determined to solve nonlinear fluid dynamics models to test this conjecture.

Even with a computer in hand, the atmospheric equations needed to be simplified to make the calculations tractable. Lorenz was more a scientist than an engineer, and more of a meteorologist than a forecaster. He did not hesitate to make simplifying assumptions if they retained the correct phenomenological behavior, even if they no longer allowed for accurate weather predictions.

He had simplified the number of atmospheric equations down to twelve. Progress was good, and by 1961, he had completed a large initial numerical study. He focused on nonperiodic solutions, which he suspected would deviate significantly from the predictions made by linear regression, and this hunch was vindicated by his numerical output. One day, as he was testing his results, he decided to save time by starting the computations midway by using mid-point results from a previous run as initial conditions. He typed in the three-digit numbers from a paper printout and went down the hall for a cup of coffee. When he returned, he looked at the printout of the twelve variables and was disappointed to find that they were not related to the previous full-time run. He immediately suspected a faulty vacuum tube, as often happened. But as he looked closer at the numbers, he realized that, at first, they tracked very well with the original run, but then began to diverge more and more rapidly until they lost all connection with the first-run numbers. His initial conditions were correct to a part in a thousand, but this small error was magnified exponentially as the solution progressed.

At this point, Lorenz recalled that he “became rather excited”. He was looking at a complete breakdown of predictability in atmospheric science. If radically different behavior arose from the smallest errors, then no measurements would ever be accurate enough to be useful for long-range forecasting. At a more fundamental level, this was a break with a long-standing tradition in science and engineering that clung to the belief that small differences produced small effects. What Lorenz had discovered, instead, was that the deterministic solution to his 12 equations was exponentially sensitive to initial conditions (known today as SIC).

The Lorenz Equations

Over the following months, he was able to show that SIC was a result of the nonperiodic solutions. The more Lorenz became familiar with the behavior of his equations, the more he felt that the 12-dimensional trajectories had a repeatable shape. He tried to visualize this shape, to get a sense of its character, but it is difficult to visualize things in twelve dimensions, and progress was slow. Then Lorenz found that when the solution was nonperiodic (the necessary condition for SIC), four of the variables settled down to zero, leaving all the dynamics to the remaining three variables.

Lorenz narrowed the equations of atmospheric instability down to three variables: the stream function, the change in temperature and the deviation in linear temperature. The only parameter in the stream function is something known as the Prandtl Number. This is a dimensionless number which is the ratio of the kinetic viscosity of the fluid to its thermal diffusion coefficient and is a physical property of the fluid. The only parameter in the change in temperature is the Rayleigh Number which is a dimensionless parameter proportional to the difference in temperature between the top and the bottom of the fluid layer. The final parameter, in the equation for the deviation in linear temperature, is the ratio of the height of the fluid layer to the width of the convection rolls. The final simplified model is given by the flow equations

The Butterfly

Lorenz finally had a 3-variable dynamical system that displayed chaos. Moreover, it had a three-dimensional state space that could be visualized directly. He ran his simulations, exploring the shape of the trajectories in three-dimensional state space for a wide range of initial conditions, and the trajectories did indeed always settle down to restricted regions of state space. They relaxed in all cases to a sort of surface that was elegantly warped, with wing-like patterns like a butterfly, as the state point of the system followed its dynamics through time. The attractor of the Lorenz equations was strange. Later, in 1971, David Ruelle (1935 – ), a Belgian-French mathematical physicist named this a “strange attractor”, and this name has become a standard part of the language of the theory of chaos.

The first graphical representation of the butterfly attractor is shown in Fig. 1 drawn by Lorenz for his 1963 publication.

Fig. 1 *Excerpts of the title, abstract and sections of Lorenz’ 1963 paper. His three-dimensional flow equations produce trajectories that relax onto a three-dimensional “strange attractor*“.

Using our modern plotting ability, the 3D character of the butterfly is shown in Fig. 2

*Fig. 2 Edward Lorenz’ chaotic butterfly*

A projection onto the x-y plane is shown in Fig. 3. In the full 3D state space the trajectories never overlap, but in the projection onto a 2D plane the trajectories are moving above and below each other.

Fig. 3 Projection of the butterfly onto the x-y plane centered on the origin.

The reason it is called a strange attractor is because all initial conditions relax onto the strange attractor, yet every trajectory on the strange attractor separates exponentially from neighboring trajectories, displaying the classic SIC property of chaos. So here is an elegant collection of trajectories that are certainly not just random noise, yet detailed prediction is still impossible. Deterministic chaos has significant structure, and generates beautiful patterns, without actual “randomness”.

Python Program

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Mon Apr 16 07:38:57 2018

@author: nolte
Introduction to Modern Dynamics, 2nd edition (Oxford University Press, 2019)

Lorenz model of atmospheric turbulence
"""
import numpy as np
import matplotlib as mpl

import matplotlib.colors as colors
import matplotlib.cm as cmx

from scipy import integrate
from matplotlib import cm
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.colors import cnames
from matplotlib import animation

plt.close('all')

jet = cm = plt.get_cmap('jet') 
values = range(10)
cNorm  = colors.Normalize(vmin=0, vmax=values[-1])
scalarMap = cmx.ScalarMappable(norm=cNorm, cmap=jet)

def solve_lorenz(N=12, angle=0.0, max_time=8.0, sigma=10.0, beta=8./3, rho=28.0):

    fig = plt.figure()
    ax = fig.add_axes([0, 0, 1, 1], projection='3d')
    ax.axis('off')

    # prepare the axes limits
    ax.set_xlim((-25, 25))
    ax.set_ylim((-35, 35))
    ax.set_zlim((5, 55))

    def lorenz_deriv(x_y_z, t0, sigma=sigma, beta=beta, rho=rho):
        """Compute the time-derivative of a Lorenz system."""
        x, y, z = x_y_z
        return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z]

    # Choose random starting points, uniformly distributed from -15 to 15
    np.random.seed(1)
    x0 = -10 + 20 * np.random.random((N, 3))

    # Solve for the trajectories
    t = np.linspace(0, max_time, int(500*max_time))
    x_t = np.asarray([integrate.odeint(lorenz_deriv, x0i, t)
                      for x0i in x0])

    # choose a different color for each trajectory
    # colors = plt.cm.viridis(np.linspace(0, 1, N))
    # colors = plt.cm.rainbow(np.linspace(0, 1, N))
    # colors = plt.cm.spectral(np.linspace(0, 1, N))
    colors = plt.cm.prism(np.linspace(0, 1, N))

    for i in range(N):
        x, y, z = x_t[i,:,:].T
        lines = ax.plot(x, y, z, '-', c=colors[i])
        plt.setp(lines, linewidth=1)

    ax.view_init(30, angle)
    plt.show()

    return t, x_t


t, x_t = solve_lorenz(angle=0, N=12)

plt.figure(2)
lines = plt.plot(t,x_t[1,:,0],t,x_t[1,:,1],t,x_t[1,:,2])
plt.setp(lines, linewidth=1)
lines = plt.plot(t,x_t[2,:,0],t,x_t[2,:,1],t,x_t[2,:,2])
plt.setp(lines, linewidth=1)
lines = plt.plot(t,x_t[10,:,0],t,x_t[10,:,1],t,x_t[10,:,2])
plt.setp(lines, linewidth=1)

To explore the parameter space of the Lorenz attractor, the key parameters to change are sigma (the Prandtl number), r (the Rayleigh number) and b on line 31 of the Python code.

References

[1] E. N. Lorenz, The essence of chaos (The Jessie and John Danz lectures; Jessie and John Danz lectures.). Seattle :: University of Washington Press (in English), 1993.

[2] E. N. Lorenz, “Deterministic Nonperiodic Flow,” Journal of the Atmospheric Sciences, vol. 20, no. 2, pp. 130-141, 1963 (1963)

August 24, 2020October 24, 2020 by David D. Nolte

Henri Poincaré and his Homoclinic Tangle

Will the next extinction-scale asteroid strike the Earth in our lifetime?

This existential question—the question of our continued existence on this planet—is rhetorical, because there are far too many bodies in our solar system to accurately calculate all trajectories of all asteroids.

The solar system is what is known as an N-body problem. And even the N is not well determined. The asteroid belt alone has over a million extinction-sized asteroids, and there are tens of millions of smaller ones that could still do major damage to life on Earth if they hit. To have a hope of calculating even one asteroid trajectory do we ignore planetary masses that are too small? What is too small? What if we only consider the Sun, the Earth and Jupiter? This is what Euler did in 1760, and he still had to make more assumptions.

Stability of the Solar System

Once Newton published his Principia, there was a pressing need to calculate the orbit of the Moon (see my blog post on the three-body problem). This was important for navigation, because if the daily position of the moon could be known with sufficient accuracy, then ships would have a means to determine their longitude at sea. However, the Moon, Earth and Sun are already a three-body problem, which still ignores the effects of Mars and Jupiter on the Moon’s orbit, not to mention the problem that the Earth is not a perfect sphere. Therefore, to have any hope of success, toy systems that were stripped of all their obfuscating detail were needed.

Euler investigated simplified versions of the three-body problem around 1760, treating a body attracted to two fixed centers of gravity moving in the plane, and he solved it using elliptic integrals. When the two fixed centers are viewed in a coordinate frame that is rotating with the Sun-Earth system, it can come close to capturing many of the important details of the system. In 1762 Euler tried another approach, called the restricted three-body problem, where he considered a massless Moon attracted to a massive Earth orbiting a massive Sun, again all in the plane. Euler could not find general solutions to this problem, but he did stumble on an interesting special case when the three bodies remain collinear throughout their motions in a rotating reference frame.

It was not the danger of asteroids that was the main topic of interest in those days, but the question whether the Earth itself is in a stable orbit and is safe from being ejected from the Solar system. Despite steadily improving methods for calculating astronomical trajectories through the nineteenth century, this question of stability remained open.

Poincaré and the King Oscar Prize of 1889

Some years ago I wrote an article for Physics Today called “The Tangled Tale of Phase Space” that tracks the historical development of phase space. One of the chief players in that story was Henri Poincaré (1854 – 1912). Henri Poincare was the Einstein before Einstein. He was a minor celebrity and was considered to be the greatest genius of his era. The event in his early career that helped launch him to stardom was a mathematics prize announced in 1887 to honor the birthday of King Oscar II of Sweden. The challenge problem was as simple as it was profound: Prove rigorously whether the solar system is stable.

This was the old N-body problem that had so far resisted solution, but there was a sense at that time that recent mathematical advances might make the proof possible. There was even a rumor that Dirichlet had outlined such a proof, but no trace of the outline could be found in his papers after his death in 1859.

The prize competition was announced in Acta Mathematica, written by the Swedish mathematician Gösta Mittag-Leffler. It stated:

Given a system of arbitrarily many mass points that attract each according to Newton’s law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.

The timing of the prize was perfect for Poincaré who was in his early thirties and just beginning to make his mark on mathematics. He was working on the theory of dynamical systems and was developing a new viewpoint that went beyond integrating single trajectories by focusing more broadly on whole classes of solutions. The question of the stability of the solar system seemed like a good problem to use to sharpen his mathematical tools. The general problem was still too difficult, so he began with Euler’s restricted three-body problem. He made steady progress, and along the way he invented an array of new techniques for studying the general properties of dynamical systems. One of these was the Poincaré section. Another was his set of integral invariants, one of which is recognized as the conservation of volume in phase space, also known as Liouville’s theorem, although it was Ludwig Boltzmann who first derived this result (see my Physics Today article). Eventually, he believed he had proven that the restricted three-body problem was stable.

By the time Poincaré had finished is prize submission, he had invented a new field of mathematical analysis, and the judges of the prize submission recognized it. Poincaré was named the winner, and his submission was prepared for publication in the Acta. However, Mittag-Leffler was a little concerned by a technical objection that had been raised, so he forwarded the comment to Poincaré for him to look at. At first, Poincaré thought the objection could easily be overcome, but as he worked on it and delved deeper, he had a sudden attack of panic. Trajectories near a saddle point did not converge. His proof of stability was wrong!

He alerted Mittag-Leffler to stop the presses, but it was too late. The first printing had been completed and review copies had already been sent to the judges. Mittag-Leffler immediately wrote to them asking for their return while Poincaré worked nonstop to produce a corrected copy. When he had completed his reanalysis, he had discovered a divergent feature of the solution to the dynamical problem near saddle points that his recognized today as the discovery of chaos. Poincaré paid for the reprinting of his paper out of his own pocket and (almost) all of the original printing was destroyed. This embarrassing moment in the life of a great mathematician was virtually forgotten until it was brought to light by the historian Barrow-Green in 1994 [1].

Poincaré is still a popular icon in France. Here is the Poincaré cafe in Paris.

A crater on the Moon is named after Poincaré.

Chaos in the Poincaré Return Map

Despite the fact that his conclusions on the stability of the 3-body problem flipped, Poincaré’s new tools for analyzing dynamical systems earned him the prize. He did not stop at his modified prize submission but continued working on systematizing his methods, publishing New Methods in Celestial Mechanics in several volumes through the 1890’s. It was here that he fully explored what happens when a trajectory approaches a saddle point of dynamical equilibrium.

The third volume of a three-book series that grew from Poincaré’s award-winning paper

To visualize a periodic trajectory, Poincaré invented a mathematical tool called a “first-return map”, also known as a Poincaré section. It was a way of taking a higher dimensional continuous trajectory and turning it into a simple iterated discrete map. Therefore, one did not need to solve continuous differential equations, it was enough to just iterate the map. In this way, complicated periodic, or nearly periodic, behavior could be explored numerically. However, even armed with this weapon, Poincaré found that iterated maps became unstable as a trajectory that originated from a saddle point approached another equivalent saddle point. Because the dynamics are periodic, the outgoing and incoming trajectories are opposite ends of the same trajectory, repeated with 2-pi periodicity. Therefore, the saddle point is also called a homoclinic point, meaning that trajectories in the discrete map intersect with themselves. (If two different trajectories in the map intersect, that is called a heteroclinic point.) When Poincaré calculated the iterations around the homoclinic point, he discovered a wild and complicated pattern in which a trajectory intersected itself many times. Poincaré wrote:

[I]f one seeks to visualize the pattern formed by these two curves and their infinite number of intersections … these intersections form a kind of lattice work, a weave, a chain-link network of infinitely fine mesh; each of the two curves can never cross itself, but it must fold back on itself in a very complicated way so as to recross all the chain-links an infinite number of times .… One will be struck by the complexity of this figure, which I am not even attempting to draw. Nothing can give us a better idea of the intricacy of the three-body problem, and of all the problems of dynamics in general…

This was the discovery of chaos! Today we call this “lattice work” the “homoclinic tangle”. He could not draw it with the tools of his day … but we can!

Chirikov’s Standard Map

The restricted 3-body problem is a bit more complicated than is needed to illustrate Poincaré’s homoclinic tangle. A much simpler model is a discrete map called Chirikov’s Map or the Standard Map. It describes the Poincaré section of a periodically kicked oscillator that rotates or oscillates in the angular direction with an angular momentm J. The map has the simple form

in which the angular momentum in updated first, and then the angle variable is updated with the new angular momentum. When plotted on the (θ,J) plane, the standard map produces a beautiful kaleidograph of intertwined trajectories piercing the Poincaré plane, as shown in the figure below. The small points or dots are successive intersections of the higher-dimensional trajectory intersecting a plane. It is possible to trace successive points by starting very close to a saddle point (on the left) and connecting successive iterates with lines. These lines merge into the black trace in the figure that emerges along the unstable manifold of the saddle point on the left and approaches the saddle point on the right generally along the stable manifold.

Fig. Standard map for K = 0.97 at the transition to full chaos. The dark line is the trajectory of the unstable manifold emerging from the saddle point at (p,0). Note the wild oscillations as it approaches the saddle point at (3pi,0).

However, as the successive iterates approach the new saddle (which is really just the old saddle point because of periodicity) it crosses the stable manifold again and again, in ever wilder swings that diverge as it approaches the saddle point. This is just one trace. By calculating traces along all four stable and unstable manifolds and carrying them through to the saddle, a lattice work, or homoclinic tangle emerges.

Two of those traces originate from the stable manifolds, so to calculate their contributions to the homoclinic tangle, one must run these traces backwards in time using the inverse Chirikov map. This is

The four traces all intertwine at the saddle point in the figure below with a zoom in on the tangle in the next figure. This is the lattice work that Poincaré glimpsed in 1889 as he worked feverishly to correct the manuscript that won him the prize that established him as one of the preeminent mathematicians of Europe.

Fig. The homoclinic tangle caused by the folding of phase space trajectories as stable and unstable manifolds criss-cross in the Poincare map at the saddle point. This was the figure that Poincaré could not attempt to draw because of its complexity.

Fig. A zoom-in of the homoclinic tangle at the saddle point as the stable and unstable manifolds create a lattice of intersections. This is the fundamental origin of chaos and the sensitivity to initial conditions (SIC) that make forecasting almost impossible in chaotic systems.

Python Code: StandmapHom.py

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
StandmapHom.py
Created on Sun Aug  2  2020
"Introduction to Modern Dynamics" 2nd Edition (Oxford, 2019)
@author: nolte
"""

import numpy as np
from matplotlib import pyplot as plt
from numpy import linalg as LA

plt.close('all')

eps = 0.97

np.random.seed(2)

plt.figure(1)

for eloop in range(0,100):

    rlast = 2*np.pi*(0.5-np.random.random())
    thlast = 4*np.pi*np.random.random()
    
    rplot = np.zeros(shape=(200,))
    thetaplot = np.zeros(shape=(200,))
    for loop in range(0,200):
        rnew = rlast + eps*np.sin(thlast)
        thnew = np.mod(thlast+rnew,4*np.pi)
        
        thetaplot[loop] = np.mod(thnew-np.pi,4*np.pi)     
        rtemp = np.mod(rnew + np.pi,2*np.pi)
        rplot[loop] = rtemp - np.pi
  
        rlast = rnew
        thlast = thnew
        
    plt.plot(np.real(thetaplot),np.real(rplot),'o',ms=0.2)
    plt.xlim(xmin=np.pi,xmax=4*np.pi)
    plt.ylim(ymin=-2.5,ymax=2.5)
        
plt.savefig('StandMap')

K = eps
eps0 = 5e-7

J = [[1,1+K],[1,1]]
w, v = LA.eig(J)

My = w[0]
Vu = v[:,0]     # unstable manifold
Vs = v[:,1]     # stable manifold

# Plot the unstable manifold
Hr = np.zeros(shape=(100,150))
Ht = np.zeros(shape=(100,150))
for eloop in range(0,100):
    
    eps = eps0*eloop

    roldu1 = eps*Vu[0]
    thetoldu1 = eps*Vu[1]
    
    Nloop = np.ceil(-6*np.log(eps0)/np.log(eloop+2))
    flag = 1
    cnt = 0
    
    while flag==1 and cnt < Nloop:
        
        ru1 = roldu1 + K*np.sin(thetoldu1)
        thetau1 = thetoldu1 + ru1
        
        roldu1 = ru1
        thetoldu1 = thetau1
        
        if thetau1 > 4*np.pi:
            flag = 0
            
        Hr[eloop,cnt] = roldu1
        Ht[eloop,cnt] = thetoldu1 + 3*np.pi
        cnt = cnt+1
    
x = Ht[0:99,12] - 2*np.pi
x2 = 6*np.pi - x
y = Hr[0:99,12]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[5:39,15] - 2*np.pi
x2 = 6*np.pi - x
y = Hr[5:39,15]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[12:69,16] - 2*np.pi
x2 = 6*np.pi - x
y = Hr[12:69,16]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[15:89,17] - 2*np.pi
x2 = 6*np.pi - x
y = Hr[15:89,17]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[30:99,18] - 2*np.pi
x2 = 6*np.pi - x
y = Hr[30:99,18]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

# Plot the stable manifold
del Hr, Ht
Hr = np.zeros(shape=(100,150))
Ht = np.zeros(shape=(100,150))
#eps0 = 0.03
for eloop in range(0,100):
    
    eps = eps0*eloop

    roldu1 = eps*Vs[0]
    thetoldu1 = eps*Vs[1]
    
    Nloop = np.ceil(-6*np.log(eps0)/np.log(eloop+2))
    flag = 1
    cnt = 0
    
    while flag==1 and cnt < Nloop:
        
        thetau1 = thetoldu1 - roldu1
        ru1 = roldu1 - K*np.sin(thetau1)

        roldu1 = ru1
        thetoldu1 = thetau1
        
        if thetau1 > 4*np.pi:
            flag = 0
            
        Hr[eloop,cnt] = roldu1
        Ht[eloop,cnt] = thetoldu1
        cnt = cnt+1
    
x = Ht[0:79,12] + np.pi
x2 = 6*np.pi - x
y = Hr[0:79,12]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[4:39,15] + np.pi
x2 = 6*np.pi - x
y = Hr[4:39,15]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[12:69,16] + np.pi
x2 =  6*np.pi - x
y = Hr[12:69,16]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[15:89,17] + np.pi
x2 =  6*np.pi - x
y = Hr[15:89,17]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

del x,y
x = Ht[30:99,18] + np.pi
x2 =  6*np.pi - x
y = Hr[30:99,18]
y2 = -y
plt.plot(x,y,linewidth =0.75)
plt.plot(x2,y2,linewidth =0.75)

References

[1] D. D. Nolte, “The tangled tale of phase space,” Physics Today, vol. 63, no. 4, pp. 33-38, Apr (2010)

[2] M. C. Gutzwiller, “Moon-Earth-Sun: The oldest three-body problem,” Reviews of Modern Physics, vol. 70, no. 2, pp. 589-639, Apr (1998)

[3] Barrow-Green J. Oscar II’s Prize Competition and the Error in Poindare’s Memoir on the Three Body Problem. Arch Hist Exact Sci 48: 107-131, 1994.

[4] Barrow-Green J. Poincaré and the three body problem. London Mathematical Society, 1997.

[5] https://the-moon.us/wiki/Poincar%C3%A9

[6] Poincaré H and Goroff DL. New methods of celestial mechanics … Edited and introduced by Daniel L. Goroff. New York: American Institute of Physics, 1993.

May 25, 2020March 22, 2022 by David D. Nolte

Timelines in the History and Physics of Dynamics (with links to primary texts)

These timelines in the History of Dynamics are organized along the Chapters in Galileo Unbound (Oxford, 2018). The book is about the physics and history of dynamics including classical and quantum mechanics as well as general relativity and nonlinear dynamics (with a detour down evolutionary dynamics and game theory along the way). The first few chapters focus on Galileo, while the following chapters follow his legacy, as theories of motion became more abstract, eventually to encompass the evolution of species within the same theoretical framework as the orbit of photons around black holes.

Galileo: A New Scientist

Galileo Galilei was the first modern scientist, launching a new scientific method that superseded, after one and a half millennia, Aristotle’s physics. Galileo’s career began with his studies of motion at the University of Pisa that were interrupted by his move to the University of Padua and his telescopic discoveries of mountains on the moon and the moons of Jupiter. Galileo became the first rock star of science, and he used his fame to promote the ideas of Copernicus and the Sun-centered model of the solar system. But he pushed too far when he lampooned the Pope. Ironically, Galileo’s conviction for heresy and his sentence to house arrest for the remainder of his life gave him the free time to finally finish his work on the physics of motion, which he published in Two New Sciences in 1638.

1543 Copernicus dies, publishes posthumously De Revolutionibus

1564 Galileo born

1581 Enters University of Pisa

1585 Leaves Pisa without a degree

1586 Invents hydrostatic balance

1588 Receives lecturship in mathematics at Pisa

1592 Chair of mathematics at Univeristy of Padua

1595 Theory of the tides

1595 Invents military and geometric compass

1596 Le Meccaniche and the principle of horizontal inertia

1600 Bruno Giordano burned at the stake

1601 Death of Tycho Brahe

1609 Galileo constructs his first telescope, makes observations of the moon

1610 Galileo discovers 4 moons of Jupiter, Starry Messenger (Sidereus Nuncius), appointed chief philosopher and mathematician of the Duke of Tuscany, moves to Florence, observes Saturn, Venus goes through phases like the moon

1611 Galileo travels to Rome, inducted into the Lyncean Academy, name “telescope” is first used

1611 Scheiner discovers sunspots

1611 Galileo meets Barberini, a cardinal

1611 Johannes Kepler, Dioptrice

1613 Letters on sunspots published by Lincean Academy in Rome

1614 Galileo denounced from the pulpit

1615 (April) Bellarmine writes an essay against Coperinicus

1615 Galileo investigated by the Inquisition

1615 Writes Letter to Christina, but does not publish it

1615 (December) travels to Rome and stays at Tuscan embassy

1616 (January) Francesco Ingoli publishes essay against Copernicus

1616 (March) Decree against copernicanism

1616 Galileo publishes theory of tides, Galileo meets with Pope Paul V, Copernicus’ book is banned, Galileo warned not to support the Coperinican system, Galileo decides not to reply to Ingoli, Galileo proposes eclipses of Jupter’s moons to determine longitude at sea

1618 Three comets appear, Grassi gives a lecture not hostile to Galileo

1618 Galileo, through Mario Guiducci, publishes scathing attack on Grassi

1619 Jesuit Grassi (Sarsi) publishes attack on Galileo concerning 3 comets

1619 Marina Gamba dies, Galileo legitimizes his son Vinczenzio

1619 Kepler’s Laws, Epitome astronomiae Copernicanae.

1623 Barberini becomes Urban VIII, The Assayer published (response to Grassi)

1624 Galileo visits Rome and Urban VIII

1629 Birth of his grandson Galileo

1630 Death of Johanes Kepler

1632 Publication of the Dialogue Concerning the Two Chief World Systems, Galileo is indicted by the Inquisition (68 years old)

1633 (February) Travels to Rome

1633 Convicted, abjurs, house arrest in Rome, then Siena, then home to Arcetri

1638 Blind, publication of Two New Sciences

1642 Galileo dies (77 years old)

Galileo’s Trajectory

Galileo’s discovery of the law of fall and the parabolic trajectory began with early work on the physics of motion by predecessors like the Oxford Scholars, Tartaglia and the polymath Simon Stevin who dropped lead weights from the leaning tower of Delft three years before Galileo (may have) dropped lead weights from the leaning tower of Pisa. The story of how Galileo developed his ideas of motion is described in the context of his studies of balls rolling on inclined plane and the surprising accuracy he achieved without access to modern timekeeping.

1583 Galileo Notices isochronism of the pendulum

1588 Receives lecturship in mathematics at Pisa

1589 – 1592 Work on projectile motion in Pisa

1592 Chair of mathematics at Univeristy of Padua

1596 Le Meccaniche and the principle of horizontal inertia

1600 Guidobaldo shares technique of colored ball

1602 Proves isochronism of the pendulum (experimentally)

1604 First experiments on uniformly accelerated motion

1604 Wrote to Scarpi about the law of fall (s ≈ t²)

1607-1608 Identified trajectory as parabolic

1609 Velocity proportional to time

1632 Publication of the Dialogue Concerning the Two Chief World Systems, Galileo is indicted by the Inquisition (68 years old)

1636 Letter to Christina published in Augsburg in Latin and Italian

1638 Blind, publication of Two New Sciences

1641 Invented pendulum clock (in theory)

1642 Dies (77 years old)

On the Shoulders of Giants

Galileo’s parabolic trajectory launched a new approach to physics that was taken up by a new generation of scientists like Isaac Newton, Robert Hooke and Edmund Halley. The English Newtonian tradition was adopted by ambitious French iconoclasts who championed Newton over their own Descartes. Chief among these was Pierre Maupertuis, whose principle of least action was developed by Leonhard Euler and Joseph Lagrange into a rigorous new science of dynamics. Along the way, Maupertuis became embroiled in a famous dispute that entangled the King of Prussia as well as the volatile Voltaire who was mourning the death of his mistress Emilie du Chatelet, the lone female French physicist of the eighteenth century.

1644 Descartes’ vortex theory of gravitation

1662 Fermat’s principle

1669 – 1690 Huygens expands on Descartes’ vortex theory

1687 Newton’s Principia

1698 Maupertuis born

1729 Maupertuis entered University in Basel. Studied under Johann Bernoulli

1736 Euler publishes Mechanica sive motus scientia analytice exposita

1737 Maupertuis report on expedition to Lapland. Earth is oblate. Attacks Cassini.

1744 Maupertuis Principle of Least Action. Euler Principle of Least Action.

1745 Maupertuis becomes president of Berlin Academy. Paris Academy cancels his membership after a campaign against him by Cassini.

1746 Maupertuis principle of Least Action for mass

1751 Samuel König disputes Maupertuis’ priority

1756 Cassini dies. Maupertuis reinstated in the French Academy

1759 Maupertuis dies

1759 du Chatelet’s French translation of Newton’s Principia published posthumously

1760 Euler 3-body problem (two fixed centers and coplanar third body)

1760-1761 Lagrange, Variational calculus (J. L. Lagrange, “Essai d’une nouvelle méthod pour dEeterminer les maxima et lest minima des formules intégrales indéfinies,” Miscellanea Teurinensia, (1760-1761))

1762 Beginning of the reign of Catherine the Great of Russia

1763 Euler colinear 3-body problem

1765 Euler publishes Theoria motus corporum solidorum on rotational mechanics

1766 Euler returns to St. Petersburg

1766 Lagrange arrives in Berlin

1772 Lagrange equilateral 3-body problem, Essai sur le problème des trois corps, 1772, Oeuvres tome 6

1775 Beginning of the American War of Independence

1776 Adam Smith Wealth of Nations

1781 William Herschel discovers Uranus

1783 Euler dies in St. Petersburg

1787 United States Constitution written

1787 Lagrange moves from Berlin to Paris

1788 Lagrange, Méchanique analytique

1789 Beginning of the French Revolution

1799 Pierre-Simon Laplace Mécanique Céleste (1799-1825)

Geometry on My Mind

This history of modern geometry focuses on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who redefined geometry and identified the importance of curvature for physics. Vector spaces, developed by Hermann Grassmann, Giuseppe Peano and David Hilbert, are examples of the kinds of abstract new spaces that are so important for modern physics, such as Hilbert space for quantum mechanics. Fractal geometry developed by Felix Hausdorff later provided the geometric language needed to solve problems in chaos theory.

1629 Fermat described higher-dim loci

1637 Descarte’s Geometry

1649 van Schooten’s commentary on Descartes Geometry

1694 Leibniz uses word “coordinate” in its modern usage

1697 Johann Bernoulli shortest distance between two points on convex surface

1732 Euler geodesic equations for implicit surfaces

1748 Euler defines modern usage of function

1801 Gauss calculates orbit of Ceres

1807 Fourier analysis (published in 1822(

1807 Gauss arrives in Göttingen

1827 Karl Gauss establishes differential geometry of curved surfaces, Disquisitiones generales circa superficies curvas

1830 Bolyai and Lobachevsky publish on hyperbolic geometry

1834 Jacobi n-fold integrals and volumes of n-dim spheres

1836 Liouville-Sturm theorem

1838 Liouville’s theorem

Liouville-1838 Download

1841 Jacobi determinants

1843 Arthur Cayley systems of n-variables

1843 Hamilton discovers quaternions

1844 Hermann Grassman n-dim vector spaces, Die Lineale Ausdehnungslehr

1846 Julius Plücker System der Geometrie des Raumes in neuer analytischer Behandlungsweise

1848 Jacobi Vorlesungen über Dynamik

1848 “Vector” coined by Hamilton

1854 Riemann’s habilitation lecture

1861 Riemann n-dim solution of heat conduction

1868 Publication of Riemann’s Habilitation

1869 Christoffel and Lipschitz work on multiple dimensional analysis

1871 Betti refers to the n-ply of numbers as a “space”.

1871 Klein publishes on non-euclidean geometry

1872 Boltzmann distribution

Boltzmann-1872 Download

1872 Jordan Essay on the geometry of n-dimensions

1872 Felix Klein’s “Erlangen Programme”

1872 Weierstrass’ Monster

1872 Dedekind cut

1872 Cantor paper on irrational numbers

1872 Cantor meets Dedekind

1872 Lipschitz derives mechanical motion as a geodesic on a manifold

Lipschitz-1872 Download

1874 Cantor beginning of set theory

1877 Cantor one-to-one correspondence between the line and n-dimensional space

1881 Gibbs codifies vector analysis

1883 Cantor set and staircase Grundlagen einer allgemeinen Mannigfaltigkeitslehre

1884 Abbott publishes Flatland

1887 Peano vector methods in differential geometry

1890 Peano space filling curve

1891 Hilbert space filling curve

1887 Darboux vol. 2 treats dynamics as a point in d-dimensional space. Applies concepts of geodesics for trajectories.

1898 Ricci-Curbastro Lesons on the Theory of Surfaces

1902 Lebesgue integral

1904 Hilbert studies integral equations

1904 von Koch snowflake

1906 Frechet thesis on square summable sequences as infinite dimensional space

1908 Schmidt Geometry in a Function Space

1910 Brouwer proof of dimensional invariance

1913 Hilbert space named by Riesz

1914 Hilbert space used by Hausdorff

1915 Sierpinski fractal triangle

1918 Hausdorff non-integer dimensions

1918 Weyl’s book Space, Time, Matter

1918 Fatou and Julia fractals

1920 Banach space

1927 von Neumann axiomatic form of Hilbert Space

1935 Frechet full form of Hilbert Space

1967 Mandelbrot coast of Britain

1982 Mandelbrot’s book The Fractal Geometry of Nature

The Tangled Tale of Phase Space

Phase space is the central visualization tool used today to study complex systems. The chapter describes the origins of phase space with the work of Joseph Liouville and Carl Jacobi that was later refined by Ludwig Boltzmann and Rudolf Clausius in their attempts to define and explain the subtle concept of entropy. The turning point in the history of phase space was when Henri Poincaré used phase space to solve the three-body problem, uncovering chaotic behavior in his quest to answer questions on the stability of the solar system. Phase space was established as the central paradigm of statistical mechanics by JW Gibbs and Paul Ehrenfest.

1804 Jacobi born (1904 – 1851) in Potsdam

1804 Napoleon I Emperor of France

1806 William Rowan Hamilton born (1805 – 1865)

1807 Thomas Young describes “Energy” in his Course on Natural Philosophy (Vol. 1 and Vol. 2)

1808 Bethoven performs his Fifth Symphony

1809 Joseph Liouville born (1809 – 1882)

1821 Hermann Ludwig Ferdinand von Helmholtz born (1821 – 1894)

1824 Carnot published Reflections on the Motive Power of Fire

1834 Jacobi n-fold integrals and volumes of n-dim spheres

1834-1835 Hamilton publishes his principle (1834, 1835).

1836 Liouville-Sturm theorem

1837 Queen Victoria begins her reign as Queen of England

1838 Liouville develops his theorem on products of n differentials satisfying certain first-order differential equations. This becomes the classic reference to Liouville’s Theorem.

Liouville-1838 Download

1847 Helmholtz Conservation of Energy (force)

1849 Thomson makes first use of “Energy” (From reading Thomas Young’s lecture notes)

1850 Clausius establishes First law of Thermodynamics: Internal energy. Second law: Heat cannot flow unaided from cold to hot. Not explicitly stated as first and second laws

Clausius-1850 Download

1851 Thomson names Clausius’ First and Second laws of Thermodynamics

1852 Thomson describes general dissipation of the universe (“energy” used in title)

1854 Thomson defined absolute temperature. First mathematical statement of 2^nd law. Restricted to reversible processes

1854 Clausius stated Second Law of Thermodynamics as inequality

1857 Clausius constructs kinetic theory, Mean molecular speeds

1858 Clausius defines mean free path, Molecules have finite size. Clausius assumed that all molecules had the same speed

Clausius-1858 Download

1860 Maxwell publishes first paper on kinetic theory. Distribution of speeds. Derivation of gas transport properties

1865 Loschmidt size of molecules

1865 Clausius names entropy

Clausius-1865 Download

1868 Boltzmann adds (Boltzmann) factor to Maxwell distribution

1872 Boltzmann transport equation and H-theorem

1876 Loschmidt reversibility paradox

1877 Boltzmann S = k logW

1890 Poincare: Recurrence Theorem. Recurrence paradox with Second Law (1893)

1896 Zermelo criticizes Boltzmann

1896 Boltzmann posits direction of time to save his H-theorem

1898 Boltzmann Vorlesungen über Gas Theorie

1905 Boltzmann kinetic theory of matter in Encyklopädie der mathematischen Wissenschaften

Boltzmann-1905 Download

1906 Boltzmann dies

1910 Paul Hertz uses “Phase Space” (Phasenraum)

1911 Ehrenfest’s article in Encyklopädie der mathematischen Wissenschaften

1913 A. Rosenthal writes the first paper using the phrase “phasenraum”, combining the work of Boltzmann and Poincaré. “Beweis der Unmöglichkeit ergodischer Gassysteme” (Ann. D. Physik, 42, 796 (1913)

1913 Plancheral, “Beweis der Unmöglichkeit ergodischer mechanischer Systeme” (Ann. D. Physik, 42, 1061 (1913). Also uses “Phasenraum”.

The Lens of Gravity

Gravity provided the backdrop for one of the most important paradigm shifts in the history of physics. Prior to Albert Einstein’s general theory of relativity, trajectories were paths described by geometry. After the theory of general relativity, trajectories are paths caused by geometry. This chapter explains how Einstein arrived at his theory of gravity, relying on the space-time geometry of Hermann Minkowski, whose work he had originally harshly criticized. The confirmation of Einstein’s theory was one of the dramatic high points in 20^th century history of physics when Arthur Eddington journeyed to an island off the coast of Africa to observe stellar deflections during a solar eclipse. If Galileo was the first rock star of physics, then Einstein was the first worldwide rock star of science.

1697 Johann Bernoulli was first to find solution to shortest path between two points on a curved surface (1697).

1728 Euler found the geodesic equation.

1783 The pair 40 Eridani B/C was discovered by William Herschel on 31 January

1783 John Michell explains infalling object would travel faster than speed of light

1796 Laplace describes “dark stars” in Exposition du system du Monde

1827 The first orbit of a binary star computed by Félix Savary for the orbit of Xi Ursae Majoris.

1827 Gauss curvature Theoriem Egregum

1844 Bessel notices periodic displacement of Sirius with period of half a century

1844 The name “geodesic line” is attributed to Liouville.

1845 Buys Ballot used musicians with absolute pitch for the first experimental verification of the Doppler effect

1854 Riemann’s habilitationsschrift

1862 Discovery of Sirius B (a white dwarf)

1868 Darboux suggested motions in n-dimensions

1872 Lipshitz first to apply Riemannian geometry to the principle of least action.

1895 Hilbert arrives in Göttingen

1902 Minkowski arrives in Göttingen

1905 Einstein’s miracle year

1906 Poincaré describes Lorentz transformations as rotations in 4D

1907 Einstein has “happiest thought” in November

1907 Einstein’s relativity review in Jahrbuch

1908 Minkowski’s Space and Time lecture

1908 Einstein appointed to unpaid position at University of Bern

1909 Minkowski dies

1909 Einstein appointed associate professor of theoretical physics at U of Zürich

1910 40 Eridani B was discobered to be of spectral type A (white dwarf)

1910 Size and mass of Sirius B determined (heavy and small)

1911 Laue publishes first textbook on relativity theory

1911 Einstein accepts position at Prague

1911 Einstein goes to the limits of special relativity applied to gravitational fields

1912 Einstein’s two papers establish a scalar field theory of gravitation

1912 Einstein moves from Prague to ETH in Zürich in fall. Begins collaboration with Grossmann.

1913 Einstein EG paper

1914 Adams publishes spectrum of 40 Eridani B

1915 Sirius B determined to be also a low-luminosity type A white dwarf

1915 Einstein Completes paper

1916 Density of 40 Eridani B by Ernst Öpik

1916 Schwarzschild paper

1916 Einstein’s publishes theory of gravitational waves

1919 Eddington expedition to Principe

1920 Eddington paper on deflection of light by the sun

1922 Willem Luyten coins phrase “white dwarf”

1924 Eddington found a set of coordinates that eliminated the singularity at the Schwarzschild radius

1926 R. H. Fowler publishes paper on degenerate matter and composition of white dwarfs

1931 Chandrasekhar calculated the limit for collapse to white dwarf stars at 1.4MS

1933 Georges Lemaitre states the coordinate singularity was an artefact

1934 Walter Baade and Fritz Zwicky proposed the existence of the neutron star only a year after the discovery of the neutron by Sir James Chadwick.

1939 Oppenheimer and Snyder showed ultimate collapse of a 3M_S “frozen star”

1958 David Finkelstein paper

1965 Antony Hewish and Samuel Okoye discovered “an unusual source of high radio brightness temperature in the Crab Nebula”. This source turned out to be the Crab Nebula neutron star that resulted from the great supernova of 1054.

1967 Jocelyn Bell and Antony Hewish discovered regular radio pulses from CP 1919. This pulsar was later interpreted as an isolated, rotating neutron star.

1967 Wheeler’s “black hole” talk

1974 Joseph Taylor and Russell Hulse discovered the first binary pulsar, PSR B1913+16, which consists of two neutron stars (one seen as a pulsar) orbiting around their center of mass.

2015 LIGO detects gravitational waves on Sept. 14 from the merger of two black holes

2017 LIGO detects the merger of two neutron stars

On the Quantum Footpath

The concept of the trajectory of a quantum particle almost vanished in the battle between Werner Heisenberg’s matrix mechanics and Erwin Schrödinger’s wave mechanics. It took Niels Bohr and his complementarity principle of wave-particle duality to cede back some reality to quantum trajectories. However, Schrödinger and Einstein were not convinced and conceived of quantum entanglement to refute the growing acceptance of the Copenhagen Interpretation of quantum physics. Schrödinger’s cat was meant to be an absurdity, but ironically it has become a central paradigm of practical quantum computers. Quantum trajectories took on new meaning when Richard Feynman constructed quantum theory based on the principle of least action, inventing his famous Feynman Diagrams to help explain quantum electrodynamics.

1885 Balmer Theory

1897 J. J. Thomson discovered the electron

1904 Thomson plum pudding model of the atom

1911 Bohr PhD thesis filed. Studies on the electron theory of metals. Visited England.

1911 Rutherford nuclear model

1911 First Solvay conference

1911 “ultraviolet catastrophe” coined by Ehrenfest

1913 Bohr combined Rutherford’s nuclear atom with Planck’s quantum hypothesis: 1913 Bohr model

1913 Ehrenfest adiabatic hypothesis

1914-1916 Bohr at Manchester with Rutherford

1916 Bohr appointed Chair of Theoretical Physics at University of Copenhagen: a position that was made just for him

1916 Schwarzschild and Epstein introduce action-angle coordinates into quantum theory

1920 Heisenberg enters University of Munich to obtain his doctorate

1920 Bohr’s Correspondence principle: Classical physics for large quantum numbers

1921 Bohr Founded Institute of Theoretical Physics (Copenhagen)

1922-1923 Heisenberg studies with Born, Franck and Hilbert at Göttingen while Sommerfeld is in the US on sabbatical.

1923 Heisenberg Doctorate. The exam does not go well. Unable to derive the resolving power of a microscope in response to question by Wien. Becomes Born’s assistant at Göttingen.

1924 Heisenberg visits Niels Bohr in Copenhagen (and met Einstein?)

1924 Heisenberg Habilitation at Göttingen on anomalous Zeeman

1924 – 1925 Heisenberg worked with Bohr in Copenhagen, returned summer of 1925 to Göttiingen

1924 Pauli exclusion principle and state occupancy

1924 de Broglie hypothesis extended wave-particle duality to matter

1924 Bohr Predicted Halfnium (72)

1924 Kronig’s proposal for electron self spin

1924 Bose (Einstein)

1925 Heisenberg paper on quantum mechanics

1925 Dirac, reading proof from Heisenberg, recognized the analogy of noncommutativity with Poisson brackets and the correspondence with Hamiltonian mechanics.

1925 Uhlenbeck and Goudschmidt: spin

1926 Born, Heisenberg, Kramers: virtual oscillators at transition frequencies: Matrix mechanics (alternative to Bohr-Kramers-Slater 1924 model of orbits). Heisenberg was Born’s student at Göttingen.

1926 Schrödinger wave mechanics

Schrödinger-1926 Download

1927 de Broglie hypotehsis confirmed by Davisson and Germer

1927 Complementarity by Bohr: wave-particle duality “Evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects.”

1927 Heisenberg uncertainty principle (Heisenberg was in Copenhagen 1926 – 1927)

Heisenberg-1927 Download

1927 Solvay Conference in Brussels

1928 Heisenberg to University of Leipzig

1928 Dirac relativistic QM equation

1929 de Broglie Nobel Prize

1930 Solvay Conference

1932 Heisenberg Nobel Prize

1932 von Neumann operator algebra

1933 Dirac Lagrangian form of QM (basis of Feynman path integral)

1933 Schrödinger and Dirac Nobel Prize

1935 Einstein, Poldolsky and Rosen EPR paper

Einstein-1935 Download

1935 Bohr’s response to Einsteins “EPR” paradox

Bohr-1935 Download

1935 Schrodinger’s cat

SchrodingerNW35 Download

SchrodingerNW35-2 Download

SchrodingerNW35-3 Download

1939 Feynman graduates from MIT

1941 Heisenberg (head of German atomic project) visits Bohr in Copenhagen

1942 Feynman PhD at Princeton, “The Principle of Least Action in Quantum Mechanics“

1942 – 1945 Manhattan Project, Bethe-Feynman equation for fission yield

1943 Bohr escapes to Sweden in a fishing boat. Went on to England secretly.

1945 Pauli Nobel Prize

1945 Death of Feynman’s wife Arline (married 4 years)

1945 Fall, Feynman arrives at Cornell ahead of Hans Bethe

1947 Shelter Island conference: Lamb Shift, did Kramer’s give a talk suggesting that infinities could be subtracted?

1947 Fall, Dyson arrives at Cornell

1948 Pocono Manor, Pennsylvania, troubled unveiling of path integral formulation and Feynman diagrams, Schwinger’s master presentation

1948 Feynman and Dirac. Summer drive across the US with Dyson

1949 Dyson joins IAS as a postdoc, trains a cohort of theorists in Feynman’s technique

1949 Karplus and Kroll first g-factor calculation

1950 Feynman moves to Cal Tech

1965 Schwinger, Tomonaga and Feynman Nobel Prize

1967 Hans Bethe Nobel Prize

From Butterflies to Hurricanes

Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are stable. In the hands of Vladimir Arnold and Jürgen Moser, this became the KAM theory of Hamiltonian chaos. This chapter shows how KAM theory fed into topology in the hands of Stephen Smale and helped launch the new field of chaos theory. Edward Lorenz discovered chaos in numerical models of atmospheric weather and discovered the eponymous strange attractor. Mathematical aspects of chaos were further developed by Mitchell Feigenbaum studying bifurcations in the logistic map that describes population dynamics.

1760 Euler 3-body problem (two fixed centers and coplanar third body)

1763 Euler colinear 3-body problem

1772 Lagrange equilateral 3-body problem

1881-1886 Poincare memoires “Sur les courbes de ́finies par une equation differentielle”

1890 Poincare “Sur le probleme des trois corps et les equations de la dynamique”. First-return map, Poincare recurrence theorem, stable and unstable manifolds

1892 – 1899 Poincare New Methods in Celestial Mechanics

1892 Lyapunov The General Problem of the Stability of Motion

1899 Poincare homoclinic trajectory

1913 Birkhoff proves Poincaré’s last geometric theorem, a special case of the three-body problem.

1927 van der Pol and van der Mark

1937 Coarse systems, Andronov and Pontryagin

1938 Morse theory

1942 Hopf bifurcation

1945 Cartwright and Littlewood study the van der Pol equation (Radar during WWII)

1954 Kolmogorov A. N., On conservation of conditionally periodic motions for a small change in Hamilton’s function.

1960 Lorenz: 12 equations

1962 Moser On Invariant Curves of Area-Preserving Mappings of an Annulus.

1963 Arnold Small denominators and problems of the stability of motion in classical and celestial mechanics

1963 Lorenz: 3 equations

1964 Arnold diffusion

1965 Smale’s horseshoe

1969 Chirikov standard map

1971 Ruelle-Takens (Ruelle coins phrase “strange attractor”)

1972 “Butterfly Effect” given for Lorenz’ talk (by Philip Merilees)

1975 Gollub-Swinney observe route to turbulence along lines of Ruelle

1975 Yorke coins “chaos theory”

1976 Robert May writes review article of the logistic map

1977 New York conference on bifurcation theory

1987 James Gleick Chaos: Making a New Science

Darwin in the Clockworks

The preceding timelines related to the central role played by families of trajectories phase space to explain the time evolution of complex systems. These ideas are extended to explore the history and development of the theory of natural evolution by Charles Darwin. Darwin had many influences, including ideas from Thomas Malthus in the context of economic dynamics. After Darwin, the ideas of evolution matured to encompass broad topics in evolutionary dynamics and the emergence of the idea of fitness landscapes and game theory driving the origin of new species. The rise of genetics with Gregor Mendel supplied a firm foundation for molecular evolution, leading to the moleculer clock of Linus Pauling and the replicator dynamics of Richard Dawkins.

1202 Fibonacci

1766 Thomas Robert Malthus born

1776 Adam Smith The Wealth of Nations

1798 Malthus “An Essay on the Principle of Population”

1817 Ricardo Principles of Political Economy and Taxation

1838 Cournot early equilibrium theory in duopoly

1848 John Stuart Mill

1848 Karl Marx Communist Manifesto

1859 Darwin Origin of Species

1867 Karl Marx Das Kapital

1871 Darwin Descent of Man, and Selection in Relation to Sex

1871 Jevons Theory of Political Economy

1871 Menger Principles of Economics

1874 Walrus Éléments d’économie politique pure, or Elements of Pure Economics (1954)

1890 Marshall Principles of Economics

1908 Hardy constant genetic variance

1910 Brouwer fixed point theorem

1910 Alfred J. Lotka autocatylitic chemical reactions

1913 Zermelo determinancy in chess

1922 Fisher dominance ratio

1922 Fisher mutations

1925 Lotka predator-prey in biomathematics

1926 Vita Volterra published same equations independently

1927 JBS Haldane (1892—1964) mutations

1928 von Neumann proves the minimax theorem

1930 Fisher ratio of sexes

1932 Wright Adaptive Landscape

1932 Haldane The Causes of Evolution

1933 Kolmogorov Foundations of the Theory of Probability

1934 Rudolph Carnap The Logical Syntax of Language

1936 John Maynard Keynes, The General Theory of Employment, Interest and Money

1936 Kolmogorov generalized predator-prey systems

1938 Borel symmetric payoff matrix

1942 Sewall Wright Statistical Genetics and Evolution

1943 McCulloch and Pitts A Logical Calculus of Ideas Immanent in Nervous Activity

1944 von Neumann and Morgenstern Theory of Games and Economic Behavior

1950 Prisoner’s Dilemma simulated at Rand Corportation

1950 John Nash Equilibrium points in n-person games and The Bargaining Problem

1951 John Nash Non-cooperative Games

1952 McKinsey Introduction to the Theory of Games (first textbook)

1953 John Nash Two-Person Cooperative Games

1953 Watson and Crick DNA

1955 Braithwaite’s Theory of Games as a Tool for the Moral Philosopher

1961 Lewontin Evolution and the Theory of Games

1962 Patrick Moran The Statistical Processes of Evolutionary Theory

1962 Linus Pauling molecular clock

1968 Motoo Kimura neutral theory of molecular evolution

1972 Maynard Smith introduces the evolutionary stable solution (ESS)

1972 Gould and Eldridge Punctuated equilibrium

1973 Maynard Smith and Price The Logic of Animal Conflict

1973 Black Scholes

1977 Eigen and Schuster The Hypercycle

1978 Replicator equation (Taylor and Jonker)

1982 Hopfield network

1982 John Maynard Smith Evolution and the Theory of Games

1984 R. Axelrod The Evolution of Cooperation

The Measure of Life

This final topic extends the ideas of dynamics into abstract spaces of high dimension to encompass the idea of a trajectory of life. Health and disease become dynamical systems defined by all the proteins and nucleic acids that comprise the physical self. Concepts from network theory, autonomous oscillators and synchronization contribute to this viewpoint. Healthy trajectories are like stable limit cycles in phase space, but disease can knock the system trajectory into dangerous regions of health space, as doctors turn to new developments in personalized medicine try to return the individual to a healthy path. This is the ultimate generalization of Galileo’s simple parabolic trajectory.

1642 Galileo dies

1656 Huygens invents pendulum clock

1665 Huygens observes “odd kind of sympathy” in synchronized clocks

1673 Huygens publishes Horologium Oscillatorium sive de motu pendulorum

1736 Euler Seven Bridges of Königsberg

Euler-Sevenbridges Download

1845 Kirchhoff’s circuit laws

1852 Guthrie four color problem

1857 Cayley trees

1858 Hamiltonian cycles

1887 Cajal neural staining microscopy

1913 Michaelis Menten dynamics of enzymes

1924 Berger, Hans: neural oscillations (Berger invented the EEG)

1926 van der Pol dimensioness form of equation

1927 van der Pol periodic forcing

1943 McCulloch and Pits mathematical model of neural nets

1948 Wiener cybernetics

1952 Hodgkin and Huxley action potential model

1952 Turing instability model

1956 Sutherland cyclic AMP

1957 Broadbent and Hammersley bond percolation

1958 Rosenblatt perceptron

1959 Erdös and Renyi random graphs

1962 Cohen EGF discovered

1965 Sebeok coined zoosemiotics

1966 Mesarovich systems biology

1967 Winfree biological rythms and coupled oscillators

1969 Glass Moire patterns in perception

1970 Rodbell G-protein

1971 phrase “strange attractor” coined (Ruelle)

1972 phrase “signal transduction” coined (Rensing)

1975 phrase “chaos theory” coined (Yorke)

1975 Werbos backpropagation

1975 Kuramoto transition

1976 Robert May logistic map

1977 Mackey-Glass equation and dynamical disease

1982 Hopfield network

1990 Strogatz and Murillo pulse-coupled oscillators

1997 Tomita systems biology of a cell

1998 Strogatz and Watts Small World network

1999 Barabasi Scale Free networks

2000 Sequencing of the human genome

November 18, 2019October 4, 2021 by David D. Nolte

Second Edition of Introduction to Modern Dynamics (Chaos, Networks, Space and Time)

The second edition of Introduction to Modern Dynamics: Chaos, Networks, Space and Time is available from Oxford University Press and Amazon.

Most physics majors will use modern dynamics in their careers: nonlinearity, chaos, network theory, econophysics, game theory, neural nets, geodesic geometry, among many others.

The first edition of Introduction to Modern Dynamics (IMD) was an upper-division junior-level mechanics textbook at the level of Thornton and Marion (Classical Dynamics of Particles and Systems) and Taylor (Classical Mechanics). IMD helped lead an emerging trend in physics education to update the undergraduate physics curriculum. Conventional junior-level mechanics courses emphasized Lagrangian and Hamiltonian physics, but notably missing from the classic subjects are modern dynamics topics that most physics majors will use in their careers: nonlinearity, chaos, network theory, econophysics, game theory, neural nets, geodesic geometry, among many others. These are the topics at the forefront of physics that drive high-tech businesses and start-ups, which is where more than half of all physicists work. IMD introduced these modern topics to junior-level physics majors in an accessible form that allowed them to master the fundamentals to prepare them for the modern world.

The second edition (IMD2) continues that trend by expanding the chapters to include additional material and topics. It rearranges several of the introductory chapters for improved logical flow and expands them to include key conventional topics that were missing in the first edition (e.g., Lagrange undetermined multipliers and expanded examples of Lagrangian applications). It is also an opportunity to correct several typographical errors and other errata that students have identified over the past several years. The second edition also has expanded homework problems.

The goal of IMD2 is to strengthen the sections on conventional topics (that students need to master to take their GREs) to make IMD2 attractive as a mainstream physics textbook for broader adoption at the junior level, while continuing the program of updating the topics and approaches that are relevant for the roles that physicists play in the 21^st century.

(New Chapters and Sections highlighted in red.)

New Features in Second Edition:

Second Edition Chapters and Sections

Part 1 Geometric Mechanics

• Expanded development of Lagrangian dynamics

• Lagrange multipliers

• More examples of applications

• Connection to statistical mechanics through the virial theorem

• Greater emphasis on action-angle variables

• The key role of adiabatic invariants

Part 1 Geometric Mechanics

Chapter 1 Physics and Geometry

1.1 State space and dynamical flows

1.2 Coordinate representations

1.3 Coordinate transformation

1.4 Uniformly rotating frames

1.5 Rigid-body motion

Chapter 2 Lagrangian Mechanics

2.1 Calculus of variations

2.2 Lagrangian applications

2.3 Lagrange’s undetermined multipliers

2.4 Conservation laws

2.5 Central force motion

2.6 Virial Theorem

Chapter 3 Hamiltonian Dynamics and Phase Space

3.1 The Hamiltonian function

3.2 Phase space

3.3 Integrable systems and action–angle variables

3.4 Adiabatic invariants

Part 2 Nonlinear Dynamics

• New section on non-autonomous dynamics

• Entire new chapter devoted to Hamiltonian mechanics

• Added importance to Chirikov standard map

• The important KAM theory of “constrained chaos” and solar system stability

• Degeneracy in Hamiltonian chaos

• A short overview of quantum chaos

• Rational resonances and the relation to KAM theory

• Synchronized chaos

Part 2 Nonlinear Dynamics

Chapter 4 Nonlinear Dynamics and Chaos

4.1 One-variable dynamical systems

4.2 Two-variable dynamical systems

4.3 Limit cycles

4.4 Discrete iterative maps

4.5 Three-dimensional state space and chaos

4.6 Non-autonomous (driven) flows

4.7 Fractals and strange attractors

Chapter 5 Hamiltonian Chaos

5.1 Perturbed Hamiltonian systems

5.2 Nonintegrable Hamiltonian systems

5.3 The Chirikov Standard Map

5.4 KAM Theory

5.5 Degeneracy and the web map

5.6 Quantum chaos

Chapter 6 Coupled Oscillators and Synchronization

6.1 Coupled linear oscillators

6.2 Simple models of synchronization

6.3 Rational resonances

6.4 External synchronization

6.5 Synchronization of Chaos

Part 3 Complex Systems

• New emphasis on diffusion on networks

• Epidemic growth on networks

• A new section of game theory in the context of evolutionary dynamics

• A new section on general equilibrium theory in economics

Part 3 Complex Systems

Chapter 7 Network Dynamics

7.1 Network structures

7.2 Random network topologies

7.3 Synchronization on networks

7.4 Diffusion on networks

7.5 Epidemics on networks

Chapter 8 Evolutionary Dynamics

81 Population dynamics

8.2 Virus infection and immune deficiency

8.3 Replicator Dynamics

8.4 Quasi-species

8.5 Game theory and evolutionary stable solutions

Chapter 9 Neurodynamics and Neural Networks

9.1 Neuron structure and function

9.2 Neuron dynamics

9.3 Network nodes: artificial neurons

9.4 Neural network architectures

9.5 Hopfield neural network

9.6 Content-addressable (associative) memory

Chapter 10 Economic Dynamics

10.1 Microeconomics and equilibrium

10.2 Macroeconomics

10.3 Business cycles

10.4 Random walks and stock prices (optional)

Part 4 Relativity and Space–Time

• Relativistic trajectories

• Gravitational waves

Part 4 Relativity and Space–Time

Chapter 11 Metric Spaces and Geodesic Motion

11.1 Manifolds and metric tensors

11.2 Derivative of a tensor

11.3 Geodesic curves in configuration space

11.4 Geodesic motion

Chapter 12 Relativistic Dynamics

12.1 The special theory

12.2 Lorentz transformations

12.3 Metric structure of Minkowski space

12.4 Relativistic trajectories

12.5 Relativistic dynamics

12.6 Linearly accelerating frames (relativistic)

Chapter 13 The General Theory of Relativity and Gravitation

13.1 Riemann curvature tensor

13.2 The Newtonian correspondence

13.3 Einstein’s field equations

13.4 Schwarzschild space–time

13.5 Kinematic consequences of gravity

13.6 The deflection of light by gravity

13.7 The precession of Mercury’s perihelion

13.8 Orbits near a black hole

13.9 Gravitational waves

Synopsis of 2^nd Ed. Chapters

Chapter 1. Physics and Geometry (Sample Chapter)

This chapter has been rearranged relative to the 1^st edition to provide a more logical flow of the overarching concepts of geometric mechanics that guide the subsequent chapters. The central role of coordinate transformations is strengthened, as is the material on rigid-body motion with expanded examples.

Chapter 2. Lagrangian Mechanics (Sample Chapter)

Much of the structure and material is retained from the 1^st edition while adding two important sections. The section on applications of Lagrangian mechanics adds many direct examples of the use of Lagrange’s equations of motion. An additional new section covers the important topic of Lagrange’s undetermined multipliers

Chapter 3. Hamiltonian Dynamics and Phase Space (Sample Chapter)

The importance of Hamiltonian systems and dynamics merits a stand-alone chapter. The topics from the 1^st edition are expanded in this new chapter, including a new section on adiabatic invariants that plays an important role in the development of quantum theory. Some topics are de-emphasized from the 1^st edition, such as general canonical transformations and the symplectic structure of phase space, although the specific transformation to action-angle coordinates is retained and amplified.

Chapter 4. Nonlinear Dynamics and Chaos

The first part of this chapter is retained from the 1^st edition with numerous minor corrections and updates of figures. The second part of the IMD 1^st edition, treating Hamiltonian chaos, will be expanded into the new Chapter 5.

Chapter 5. Hamiltonian Chaos

This new stand-alone chapter expands on the last half of Chapter 3 of the IMD 1^st edition. The physical character of Hamiltonian chaos is substantially distinct from dissipative chaos that it deserves its own chapter. It is also a central topic of interest for complex systems that are either conservative or that have integral invariants, such as our N-body solar system that played such an important role in the history of chaos theory beginning with Poincaré. The new chapter highlights Poincaré’s homoclinic tangle, illustrated by the Chirikov Standard Map. The Standard Map is an excellent introduction to KAM theory, which is one of the crowning achievements of the theory of dynamical systems by Komogorov, Arnold and Moser, connecting to deeper aspects of synchronization and rational resonances that drive the structure of systems as diverse as the rotation of the Moon and the rings of Saturn. This is also a perfect lead-in to the next chapter on synchronization. An optional section at the end of this chapter briefly discusses quantum chaos to show how Hamiltonian chaos can be extended into the quantum regime.

Chapter 6. Synchronization

This is an updated version of the IMD 1^st ed. chapter. It has a reduced initial section on coupled linear oscillators, retaining the key ideas about linear eigenmodes but removing some irrelevant details in the 1^st edition. A new section is added that defines and emphasizes the importance of quasi-periodicity. A new section on the synchronization of chaotic oscillators is added.

Chapter 7. Network Dynamics

This chapter rearranges the structure of the chapter from the 1^st edition, moving synchronization on networks earlier to connect from the previous chapter. The section on diffusion and epidemics is moved to the back of the chapter and expanded in the 2^nd edition into two separate sections on these topics, adding new material on discrete matrix approaches to continuous dynamics.

Chapter 8. Neurodynamics and Neural Networks

This chapter is retained from the 1^st edition with numerous minor corrections and updates of figures.

Chapter 9. Evolutionary Dynamics

Two new sections are added to this chapter. A section on game theory and evolutionary stable solutions introduces core concepts of evolutionary dynamics that merge well with the other topics of the chapter such as the pay-off matrix and replicator dynamics. A new section on nearly neutral networks introduces new types of behavior that occur in high-dimensional spaces which are counter intuitive but important for understanding evolutionary drift.

Chapter 10. Economic Dynamics

This chapter will be significantly updated relative to the 1^st edition. Most of the sections will be rewritten with improved examples and figures. Three new sections will be added. The 1^st edition section on consumer market competition will be split into two new sections describing the Cournot duopoly and Pareto optimality in one section, and Walras’ Law and general equilibrium theory in another section. The concept of the Pareto frontier in economics is becoming an important part of biophysical approaches to population dynamics. In addition, new trends in economics are drawing from general equilibrium theory, first introduced by Walras in the nineteenth century, but now merging with modern ideas of fixed points and stable and unstable manifolds. A third new section is added on econophysics, highlighting the distinctions that contrast economic dynamics (phase space dynamical approaches to economics) from the emerging field of econophysics (statistical mechanics approaches to economics).

Chapter 11. Metric Spaces and Geodesic Motion

This chapter is retained from the 1^st edition with several minor corrections and updates of figures.

Chapter 12. Relativistic Dynamics

This chapter is retained from the 1^st edition with minor corrections and updates of figures. More examples will be added, such as invariant mass reconstruction. The connection between relativistic acceleration and Einstein’s equivalence principle will be strengthened.

Chapter 13. The General Theory of Relativity and Gravitation

This chapter is retained from the 1^st edition with minor corrections and updates of figures. A new section will derive the properties of gravitational waves, given the spectacular success of LIGO and the new field of gravitational astronomy.

Homework Problems:

All chapters will have expanded and updated homework problems. Many of the homework problems from the 1^st edition will remain, but the number of problems at the end of each chapter will be nearly doubled, while removing some of the less interesting or problematic problems.

Bibliography

D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd Ed. (Oxford University Press, 2019)

Phase Space

Quantum Phase Space: The Wigner Distribution Function

The Harmonic Oscillator

Quantum Pendulum and Separatrix Chaos

Quantum Double-Well and Separatrix Chaos

Conclusion

YouTube Video

References

Disaster in the Poconos

Coherent States

Quantum Trajectories

Youtube Video

YouTube Video

References

The Quantum Mechanics of Classically Chaotic Systems

The Quantum Circus

Quantum Chaos versus Laser Speckle

YouTube Video

References

The Three-Body Problem

Resonance

Matlab Code: body3.m

References:

Bifurcation Physics

Sliding Mass on a Rotating Hoop

Golden Behavior

Stein’s Gate and the Divergence Meter

The Local Basin of Attraction

Python Program: SteinsGate2D.py

The Lorenz Butterfly

Discussion and Extensions

References

Lorenz’ Butterfly

The Lorenz Equations

The Butterfly

Python Program

References

Stability of the Solar System

Poincaré and the King Oscar Prize of 1889

Chaos in the Poincaré Return Map

Chirikov’s Standard Map

Python Code: StandmapHom.py

References

Galileo: A New Scientist

Galileo’s Trajectory

On the Shoulders of Giants

Geometry on My Mind

The Tangled Tale of Phase Space

The Lens of Gravity

On the Quantum Footpath

From Butterflies to Hurricanes

Darwin in the Clockworks

The Measure of Life

New Features in Second Edition:

Second Edition Chapters and Sections

Synopsis of 2nd Ed. Chapters

Bibliography

Synopsis of 2^nd Ed. Chapters