The Iconic Eikonal and the Optical Path

Nature loves the path of steepest descent.  Place a ball on a smooth curved surface and release it, and it will instantansouly accelerate in the direction of steepest descent.  Shoot a laser beam from an oblique angle onto a piece of glass to hit a target inside, and the path taken by the beam is the only path that decreases the distance to the target in the shortest time.  Diffract a stream of electrons from the surface of a crystal, and quantum detection events are greatest at the positions where the troughs and peaks of the deBroglie waves converge the most.  The first example is Newton’s second law.  The second example is Fermat’s principle.  The third example is Feynman’s path-integral formulation of quantum mechanics.  They all share in common a minimization principle—the principle of least action—that the path of a dynamical system is the one that minimizes a property known as “action”.

The Eikonal Equation is the “F = ma” of ray optics.  It’s solutions describe the paths of light rays through complicated media.

         The principle of least action, first proposed by the French physicist Maupertuis through mechanical analogy, became a principle of Lagrangian mechanics in the hands of Lagrange, but was still restricted to mechanical systems of particles.  The principle was generalized forty years later by Hamilton, who began by considering the propagation of light waves, and ended by transforming mechanics into a study of pure geometry divorced from forces and inertia.  Optics played a key role in the development of mechanics, and mechanics returned the favor by giving optics the Eikonal Equation.  The Eikonal Equation is the “F = ma” of ray optics.  It’s solutions describe the paths of light rays through complicated media.

Malus’ Theorem

Anyone who has taken a course in optics knows that Étienne-Louis Malus (1775-1812) discovered the polarization of light, but little else is taught about this French mathematician who was one of the savants Napoleon had taken along with himself when he invaded Egypt in 1798.  After experiencing numerous horrors of war and plague, Malus returned to France damaged but wiser.  He discovered the polarization of light in the Fall of 1808 as he was playing with crystals of icelandic spar at sunset and happened to view last rays of the sun reflected from the windows of the Luxumbourg palace.  Icelandic spar produces double images in natural light because it is birefringent.  Malus discovered that he could extinguish one of the double images of the Luxumbourg windows by rotating the crystal a certain way, demonstrating that light is polarized by reflection.  The degree to which light is extinguished as a function of the angle of the polarizing crystal is known as Malus’ Law

Fronts-piece to the Description de l’Égypte , the first volume published by Joseph Fourier in 1808 based on the report of the savants of L’Institute de l’Égypte that included Monge, Fourier and Malus, among many other French scientists and engineers.

         Malus had picked up an interest in the general properties of light and imaging during lulls in his ordeal in Egypt.  He was an emissionist following his compatriot Laplace, rather than an undulationist following Thomas Young.  It is ironic that the French scientists were staunchly supporting Newton on the nature of light, while the British scientist Thomas Young was trying to upend Netwonian optics.  Almost all physicists at that time were emissionists, only a few years after Young’s double-slit experiment of 1804, and few serious scientists accepted Young’s theory of the wave nature of light until Fresnel and Arago supplied the rigorous theory and experimental proofs much later in 1819. 

Malus’ Theorem states that rays perpendicular to an initial surface are perpendicular to a later surface after reflection in an optical system. This theorem is the starting point for the Eikonal ray equation, as well as for modern applications in adaptive optics. This figure shows a propagating aberrated wavefront that is “compensated” by a deformable mirror to produce a tight focus.

         As a prelude to his later discovery of polarization, Malus had earlier proven a theorem about trajectories that particles of light take through an optical system.  One of the key questions about the particles of light in an optical system was how they formed images.  The physics of light particles moving through lenses was too complex to treat at that time, but reflection was relatively easy based on the simple reflection law.  Malus proved a theorem mathematically that after a reflection from a curved mirror, a set of rays perpendicular to an initial nonplanar surface would remain perpendicular at a later surface after reflection (this property is closely related to the conservation of optical etendue).  This is known as Malus’ Theorem, and he thought it only held true after a single reflection, but later mathematicians proved that it remains true even after an arbitrary number of reflections, even in cases when the rays intersect to form an optical effect known as a caustic.  The mathematics of caustics would catch the interest of an Irish mathematician and physicist who helped launch a new field of mathematical physics.

Etienne-Louis Malus

Hamilton’s Characteristic Function

William Rowan Hamilton (1805 – 1865) was a child prodigy who taught himself thirteen languages by the time he was thirteen years old (with the help of his linguist uncle), but mathematics became his primary focus at Trinity College at the University in Dublin.  His mathematical prowess was so great that he was made the Astronomer Royal of Ireland while still an undergraduate student.  He also became fascinated in the theory of envelopes of curves and in particular to the mathematics of caustic curves in optics. 

         In 1823 at the age of 18, he wrote a paper titled Caustics that was read to the Royal Irish Academy.  In this paper, Hamilton gave an exceedingly simple proof of Malus’ Law, but that was perhaps the simplest part of the paper.  Other aspects were mathematically obscure and reviewers requested further additions and refinements before publication.  Over the next four years, as Hamilton expanded this work on optics, he developed a new theory of optics, the first part of which was published as Theory of Systems of Rays in 1827 with two following supplements completed by 1833 but never published.

         Hamilton’s most important contribution to optical theory (and eventually to mechanics) he called his characteristic function.  By applying the principle of Fermat’s least time, which he called his principle of stationary action, he sought to find a single unique function that characterized every path through an optical system.  By first proving Malus’ Theorem and then applying the theorem to any system of rays using the principle of stationary action, he was able to construct two partial differential equations whose solution, if it could be found, defined every ray through the optical system.  This result was completely general and could be extended to include curved rays passing through inhomogeneous media.  Because it mapped input rays to output rays, it was the most general characterization of any defined optical system.  The characteristic function defined surfaces of constant action whose normal vectors were the rays of the optical system.  Today these surfaces of constant action are called the Eikonal function (but how it got its name is the next chapter of this story).  Using his characteristic function, Hamilton predicted a phenomenon known as conical refraction in 1832, which was subsequently observed, launching him to a level of fame unusual for an academic.

         Once Hamilton had established his principle of stationary action of curved light rays, it was an easy step to extend it to apply to mechanical systems of particles with curved trajectories.  This step produced his most famous work On a General Method in Dynamics published in two parts in 1834 and 1835 [1] in which he developed what became known as Hamiltonian dynamics.  As his mechanical work was extended by others including Jacobi, Darboux and Poincaré, Hamilton’s work on optics was overshadowed, overlooked and eventually lost.  It was rediscovered when Schrödinger, in his famous paper of 1926, invoked Hamilton’s optical work as a direct example of the wave-particle duality of quantum mechanics [2]. Yet in the interim, a German mathematician tackled the same optical problems that Hamilton had seventy years earlier, and gave the Eikonal Equation its name.

Bruns’ Eikonal

The German mathematician Heinrich Bruns (1848-1919) was engaged chiefly with the measurement of the Earth, or geodesy.  He was a professor of mathematics in Berlin and later Leipzig.  One claim fame was that one of his graduate students was Felix Hausdorff [3] who would go on to much greater fame in the field of set theory and measure theory (the Hausdorff dimension was a precursor to the fractal dimension).  Possibly motivated by his studies done with Hausdorff on refraction of light by the atmosphere, Bruns became interested in Malus’ Theorem for the same reasons and with the same goals as Hamilton, yet was unaware of Hamilton’s work in optics. 

         The mathematical process of creating “images”, in the sense of a mathematical mapping, made Bruns think of the Greek word  eikwn which literally means “icon” or “image”, and he published a small book in 1895 with the title Das Eikonal in which he derived a general equation for the path of rays through an optical system.  His approach was heavily geometrical and is not easily recognized as an equation arising from variational principals.  It rediscovered most of the results of Hamilton’s paper on the Theory of Systems of Rays and was thus not groundbreaking in the sense of new discovery.  But it did reintroduce the world to the problem of systems of rays, and his name of Eikonal for the equations of the ray paths stuck, and was used with increasing frequency in subsequent years.  Arnold Sommerfeld (1868 – 1951) was one of the early proponents of the Eikonal equation and recognized its connection with action principles in mechanics. He discussed the Eikonal equation in a 1911 optics paper with Runge [4] and in 1916 used action principles to extend Bohr’s model of the hydrogen atom [5]. While the Eikonal approach was not used often, it became popular in the 1960’s when computational optics made numerical solutions possible.

Lagrangian Dynamics of Light Rays

In physical optics, one of the most important properties of a ray passing through an optical system is known as the optical path length (OPL).  The OPL is the central quantity that is used in problems of interferometry, and it is the central property that appears in Fermat’s principle that leads to Snell’s Law.  The OPL played an important role in the history of the calculus when Johann Bernoulli in 1697 used it to derive the path taken by a light ray as an analogy of a brachistochrone curve – the curve of least time taken by a particle between two points.

            The OPL between two points in a refractive medium is the sum of the piecewise product of the refractive index n with infinitesimal elements of the path length ds.  In integral form, this is expressed as

where the “dot” is a derivative with respedt to s.  The optical Lagrangian is recognized as

The Lagrangian is inserted into the Euler equations to yield (after some algebra, see Introduction to Modern Dynamics pg. 336)

This is a second-order ordinary differential equation in the variables xa that define the ray path through the system.  It is literally a “trajectory” of the ray, and the Eikonal equation becomes the F = ma of ray optics.

Hamiltonian Optics

In a paraxial system (in which the rays never make large angles relative to the optic axis) it is common to select the position z as a single parameter to define the curve of the ray path so that the trajectory is parameterized as

where the derivatives are with respect to z, and the effective Lagrangian is recognized as

The Hamiltonian formulation is derived from the Lagrangian by defining an optical Hamiltonian as the Legendre transform of the Lagrangian.  To start, the Lagrangian is expressed in terms of the generalized coordinates and momenta.  The generalized optical momenta are defined as

This relationship leads to an alternative expression for the Eikonal equation (also known as the scalar Eikonal equation) expressed as

where S(x,y,z) = const. is the eikonal function.  The  momentum vectors are perpendicular to the surfaces of constant S, which are recognized as the wavefronts of a propagating wave.

            The Lagrangian can be restated as a function of the generalized momenta as

and the Legendre transform that takes the Lagrangian into the Hamiltonian is

The trajectory of the rays is the solution to Hamilton’s equations of motion applied to this Hamiltonian

Light Orbits

If the optical rays are restricted to the x-y plane, then Hamilton’s equations of motion can be expressed relative to the path length ds, and the momenta are pa = ndxa/ds.  The ray equations are (simply expressing the 2 second-order Eikonal equation as 4 first-order equations)

where the dot is a derivative with respect to the element ds.

As an example, consider a radial refractive index profile in the x-y plane

where r is the radius on the x-y plane. Putting this refractive index profile into the Eikonal equations creates a two-dimensional orbit in the x-y plane. The following Python code solves for individual trajectories.

Python Code: raysimple.py

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Tue May 28 11:50:24 2019

@author: nolte
"""

import numpy as np
import matplotlib as mpl
from mpl_toolkits.mplot3d import Axes3D
from scipy import integrate
from matplotlib import pyplot as plt
from matplotlib import cm
import time
import os

plt.close('all')

# selection 1 = Gaussian
# selection 2 = Donut
selection = 1

print(' ')
print('raysimple.py')

def refindex(x,y):
    
    if selection == 1:
        
        sig = 10
        
        n = 1 + np.exp(-(x**2 + y**2)/2/sig**2)
        nx = (-2*x/2/sig**2)*np.exp(-(x**2 + y**2)/2/sig**2)
        ny = (-2*y/2/sig**2)*np.exp(-(x**2 + y**2)/2/sig**2)
        
    elif selection == 2:
        
        sig = 10;
        r2 = (x**2 + y**2)
        r1 = np.sqrt(r2)
        np.expon = np.exp(-r2/2/sig**2)
        
        n = 1+0.3*r1*np.expon;
        nx = 0.3*r1*(-2*x/2/sig**2)*np.expon + 0.3*np.expon*2*x/r1
        ny = 0.3*r1*(-2*y/2/sig**2)*np.expon + 0.3*np.expon*2*y/r1
    
        
    return [n,nx,ny]


def flow_deriv(x_y_z,tspan):
    x, y, z, w = x_y_z
    
    n, nx, ny = refindex(x,y)
    
    yp = np.zeros(shape=(4,))
    yp[0] = z/n
    yp[1] = w/n
    yp[2] = nx
    yp[3] = ny
    
    return yp
                
V = np.zeros(shape=(100,100))
for xloop in range(100):
    xx = -20 + 40*xloop/100
    for yloop in range(100):
        yy = -20 + 40*yloop/100
        n,nx,ny = refindex(xx,yy) 
        V[yloop,xloop] = n

fig = plt.figure(1)
contr = plt.contourf(V,100, cmap=cm.coolwarm, vmin = 1, vmax = 3)
fig.colorbar(contr, shrink=0.5, aspect=5)    
fig = plt.show()


v1 = 0.707      # Change this initial condition
v2 = np.sqrt(1-v1**2)
y0 = [12, v1, 0, v2]     # Change these initial conditions

tspan = np.linspace(1,1700,1700)

y = integrate.odeint(flow_deriv, y0, tspan)

plt.figure(2)
lines = plt.plot(y[1:1550,0],y[1:1550,1])
plt.setp(lines, linewidth=0.5)
plt.show()
Gaussian refractive index profile in the x-y plane. From raysimple.py.
Ray orbits around the center of the Gaussian refractive index profile. From raysimple.py

Bibliography

An excellent textbook on geometric optics from Hamilton’s point of view is K. B. Wolf, Geometric Optics in Phase Space (Springer, 2004). Another is H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1992).

A rather older textbook on geometrical optics is by J. L. Synge, Geometrical Optics: An Introduction to Hamilton’s Method (Cambridge University Press, 1962) showing the derivation of the ray equations in the final chapter using variational methods. Synge takes a dim view of Bruns’ term “Eikonal” since Hamilton got there first and Bruns was unaware of it.

A book that makes an especially strong case for the Optical-Mechanical analogy of Fermat’s principle, connecting the trajectories of mechanics to the paths of optical rays is Daryl Holm, Geometric Mechanics: Part I Dynamics and Symmetry (Imperial College Press 2008).

The Eikonal ray equation is derived from the geodesic equation (or rather as a geodesic equation) in D. D. Nolte, Introduction to Modern Dynamics (Oxford, 2015).


References

[1] Hamilton, W. R. “On a general method in dynamics I.” Mathematical Papers, I ,103-161: 247-308. (1834); Hamilton, W. R. “On a general method in dynamics II.” Mathematical Papers, I ,103-161: 95-144. (1835)

[2] Schrodinger, E. “Quantification of the eigen-value problem.” Annalen Der Physik 79(6): 489-527. (1926)

[3] For the fateful story of Felix Hausdorff (aka Paul Mongré) see Chapter 9 of Galileo Unbound (Oxford, 2018).

[4] Sommerfeld, A. and J. Runge. “The application of vector calculations on the basis of geometric optics.” Annalen Der Physik 35(7): 277-298. (1911)

[5] Sommerfeld, A. “The quantum theory of spectral lines.” Annalen Der Physik 51(17): 1-94. (1916)


Biased Double-well Potential: Bistability, Bifurcation and Hysteresis

Bistability, bifurcation and hysteresis are ubiquitous phenomena that arise from nonlinear dynamics and have considerable importance for technology applications.  For instance, the hysteresis associated with the flipping of magnetic domains under magnetic fields is the central mechanism for magnetic memory, and bistability is a key feature of switching technology.

… one of the most commonly encountered bifurcations is called a saddle-node bifurcation, which is the bifurcation that occurs in the biased double-well potential.

One of the simplest models for bistability and hysteresis is the one-dimensional double-well potential biased by a changing linear potential.  An example of a double-well potential with a bias is

where the parameter c is a control parameter (bias) that can be adjusted or that changes slowly in time c(t).  This dynamical system is also known as the Duffing oscillator. The net double-well potentials for several values of the control parameter c are shown in Fig. 1.   With no bias, there are two degenerate energy minima.  As c is made negative, the left well has the lowest energy, and as c is made positive the right well has the lowest energy.

The dynamics of this potential energy profile can be understood by imagining a small ball that responds to the local forces exerted by the potential.  For large negative values of c the ball will have its minimum energy in the left well.  As c is increased, the energy of the left well increases, and rises above the energy of the right well.  If the ball began in the left well, even when the left well has a higher energy than the right, there is a potential barrier that the ball cannot overcome and it remains on the left.  This local minimum is a stable equilibrium, but it is called “metastable” because it is not a global minimum of the system.  Metastability is the origin of hysteresis.

Fig. 1 A biased double-well potential in one dimension. The thresholds to destroy the local metastable minima are c = +/-1.05. For values beyond threshold, only a single minimum exists with no barrier. Hysteresis is caused by the mass being stuck in the metastable (upper) minimum because it has insufficient energy to overcome the potential barrier, until the barrier disappears at threshold and the ball rolls all the way down to the bottom to the new location. When the bias is slowly reversed, the new location becomes metastable, until the ball can overcome the barrier and roll down to its original minimum, etc.

           Once sufficient bias is applied that the local minimum disappears, the ball will roll downhill to the new minimum on the right, and in the presence of dissipation, it will come to rest in the new minimum.  The bias can then be slowly lowered, reversing this process. Because of the potential barrier, the bias must change sign and be strong enough to remove the stability of the now metastable fixed point with the ball on the right, allowing the ball to roll back down to its original location on the left.  This “overshoot” defines the extent of the hysteresis. The fact that there are two minima, and that one is metastable with a barrier between the two, produces “bistability”, meaning that there are two stable fixed points for the same control parameter.

           For illustration, assume a mass obeys the flow equation

including a damping term, where the force is the negative gradient of the potential energy.  The bias parameter c can be time dependent, beginning beyond the negative threshold and slowly increasing until it exceeds the positive threshold, and then reversing and decreasing again.  The position of the mass is locally a damped oscillator until a threshold is passed, and then the mass falls into the global minimum, as shown in Fig. 2. As the bias is reversed, it remains in the metastable minimum on the right until the control parameter passes threshold, and then the mass drops into the left minimum that is now a global minimum.

Fig. 2 Hysteresis diagram. The mass begins in the left well. As the parameter c increases, the mass remains in the well, even though it is no longer the global minimum when c becomes positive. When c passes the positive threshold (around 1.05 for this example), the mass falls into the right well, with damped oscillation. Then the control parameter c is decreased slowly until the negative threshold is passed, and the mass switches to the left well with damped oscillations. The difference between the “switch up” and “switch down” values of the control parameter represents the “hysteresis” of the this system.

The sudden switching of the biased double-well potential represents what is known as a “bifurcation”. A bifurcation is a sudden change in the qualitative behavior of a system caused by a small change in a control variable. Usually, a bifurcation occurs when the number of attractors of a system changes. There is a fairly large menagerie of different types of bifurcations, but one of the most commonly encountered bifurcations is called a saddle-node bifurcation, which is the bifurcation that occurs in the biased double-well potential. In fact, there are two saddle-node bifurcations.

Bifurcations are easily portrayed by creating a joint space between phase space and the one (or more) control parameters that induce the bifurcation. The phase space of the double well is two dimensional (position, velocity) with three fixed points, but the change in the number of fixed points can be captured by taking a projection of the phase space onto a lower-dimensional manifold. In this case, the projection is simply along the x-axis. Therefore a “co-dimensional phase space” can be constructed with the x-axis as one dimension and the control parameter as the other. This is illustrated in Fig. 3. The cubic curve traces out the solutions to the fixed-point equation

For a given value of the control parameter c there are either three solutions or one solution. The values of c where the number of solutions changes discontinuously is the bifurcation point c*. Two examples of the potential function are shown on the right for c = +1 and c = -0.5 showing the locations of the three fixed points.

Fig. 3 The co-dimension phase space combines the one-dimensional dynamics along the position x with the control parameter. For a given value of c, there are three or one solution for the fixed point. When there are three solutions, two are stable (the double minima) and one is unstable (the saddle). As the magnitude of the bias increases, one stable node annihilates with the unstable node (a minimum and the saddle merge) and the dynamics “switch” to the other minimum.

The threshold value in this example is c* = 1.05. When |c| < c* the two stable fixed points are the two minima of the double-well potential, and the unstable fixed point is the saddle between them. When |c| > c* then the single stable fixed point is the single minimum of the potential function. The saddle-node bifurcation takes its name from the fact (illustrated here) that the unstable fixed point is a saddle, and at the bifurcation the saddle point annihilates with one of the stable fixed points.

The following Python code illustrates the behavior of a biased double-well potential, with damping, in which the control parameter changes slowly with a sinusoidal time dependence.

Python Code

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Wed Apr 17 15:53:42 2019

@author: nolte
"""

import numpy as np
from scipy import integrate
from scipy import signal
from matplotlib import pyplot as plt

plt.close('all')
T = 400
Amp = 3.5

def solve_flow(y0,c0,lim = [-3,3,-3,3]):

    def flow_deriv(x_y, t, c0):
        #"""Compute the time-derivative of a Medio system."""
        x, y = x_y

        return [y,-0.5*y - x**3 + 2*x + x*(2*np.pi/T)*Amp*np.cos(2*np.pi*t/T) + Amp*np.sin(2*np.pi*t/T)]

    tsettle = np.linspace(0,T,101)   
    yinit = y0;
    x_tsettle = integrate.odeint(flow_deriv,yinit,tsettle,args=(T,))
    y0 = x_tsettle[100,:]
    
    t = np.linspace(0, 1.5*T, 2001)
    x_t = integrate.odeint(flow_deriv, y0, t, args=(T,))
    c  = Amp*np.sin(2*np.pi*t/T)
        
    return t, x_t, c

eps = 0.0001

for loop in range(0,100):
    c = -1.2 + 2.4*loop/100 + eps;
    xc[loop]=c
    
    coeff = [-1, 0, 2, c]
    y = np.roots(coeff)
    
    xtmp = np.real(y[0])
    ytmp = np.real(y[1])
    
    X[loop] = np.min([xtmp,ytmp])
    Y[loop] = np.max([xtmp,ytmp])
    Z[loop]= np.real(y[2])


plt.figure(1)
lines = plt.plot(xc,X,xc,Y,xc,Z)
plt.setp(lines, linewidth=0.5)
plt.show()
plt.title('Roots')

y0 = [1.9, 0]
c0 = -2.

t, x_t, c = solve_flow(y0,c0)
y1 = x_t[:,0]
y2 = x_t[:,1]

plt.figure(2)
lines = plt.plot(t,y1)
plt.setp(lines, linewidth=0.5)
plt.show()
plt.ylabel('X Position')
plt.xlabel('Time')

plt.figure(3)
lines = plt.plot(c,y1)
plt.setp(lines, linewidth=0.5)
plt.show()
plt.ylabel('X Position')
plt.xlabel('Control Parameter')
plt.title('Hysteresis Figure')

Further Reading:

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

The Pendulum Lab

The Wonderful World of Hamiltonian Maps

Hamiltonian systems are freaks of nature.  Unlike the everyday world we experience that is full of dissipation and inefficiency, Hamiltonian systems live in a world free of loss.  Despite how rare this situation is for us, this unnatural state happens commonly in two extremes: orbital mechanics and quantum mechanics.  In the case of orbital mechanics, dissipation does exist, most commonly in tidal effects, but effects of dissipation in the orbits of moons and planets takes eons to accumulate, making these systems effectively free of dissipation on shorter time scales.  Quantum mechanics is strictly free of dissipation, but there is a strong caveat: ALL quantum states need to be included in the quantum description.  This includes the coupling of discrete quantum states to their environment.  Although it is possible to isolate quantum systems to a large degree, it is never possible to isolate them completely, and they do interact with the quantum states of their environment, if even just the black-body radiation from their container, and even if that container is cooled to milliKelvins.  Such interactions involve so many degrees of freedom, that it all behaves like dissipation.  The origin of quantum decoherence, which poses such a challenge for practical quantum computers, is the entanglement of quantum systems with their environment.

Liouville’s theorem plays a central role in the explanation of the entropy and ergodic properties of ideal gases, as well as in Hamiltonian chaos.

Liouville’s Theorem and Phase Space

A middle ground of practically ideal Hamiltonian mechanics can be found in the dynamics of ideal gases. This is the arena where Maxwell and Boltzmann first developed their theories of statistical mechanics using Hamiltonian physics to describe the large numbers of particles.  Boltzmann applied a result he learned from Jacobi’s Principle of the Last Multiplier to show that a volume of phase space is conserved despite the large number of degrees of freedom and the large number of collisions that take place.  This was the first derivation of what is today known as Liouville’s theorem.

Close-up of the Lozi Map with B = -1 and C = 0.5.

In 1838 Joseph Liouville, a pure mathematician, was interested in classes of solutions of differential equations.  In a short paper, he showed that for one class of differential equation one could define a property that remained invariant under the time evolution of the system.  This purely mathematical paper by Liouville was expanded upon by Jacobi, who was a major commentator on Hamilton’s new theory of dynamics, contributing much of the mathematical structure that we associate today with Hamiltonian mechanics.  Jacobi recognized that Hamilton’s equations were of the same class as the ones studied by Liouville, and the conserved property was a product of differentials.  In the mid-1800’s the language of multidimensional spaces had yet to be invented, so Jacobi did not recognize the conserved quantity as a volume element, nor the space within which the dynamics occurred as a space.  Boltzmann recognized both, and he was the first to establish the principle of conservation of phase space volume. He named this principle after Liouville, even though it was actually Boltzmann himself who found its natural place within the physics of Hamiltonian systems [1].

Liouville’s theorem plays a central role in the explanation of the entropy of ideal gases, as well as in Hamiltonian chaos.  In a system with numerous degrees of freedom, a small volume of initial conditions is stretched and folded by the dynamical equations as the system evolves.  The stretching and folding is like what happens to dough in a bakers hands.  The volume of the dough never changes, but after a long time, a small spot of food coloring will eventually be as close to any part of the dough as you wish.  This analogy is part of the motivation for ergodic systems, and this kind of mixing is characteristic of Hamiltonian systems, in which trajectories can diffuse throughout the phase space volume … usually.

Interestingly, when the number of degrees of freedom are not so large, there is a middle ground of Hamiltonian systems for which some initial conditions can lead to chaotic trajectories, while other initial conditions can produce completely regular behavior.  For the right kind of systems, the regular behavior can hem in the irregular behavior, restricting it to finite regions.  This was a major finding of the KAM theory [2], named after Kolmogorov, Arnold and Moser, which helped explain the regions of regular motion separating regions of chaotic motion as illustrated in Chirikov’s Standard Map.

Discrete Maps

Hamilton’s equations are ordinary continuous differential equations that define a Hamiltonian flow in phase space.  These equations can be solved using standard techniques, such as Runge-Kutta.  However, a much simpler approach for exploring Hamiltonian chaos uses discrete maps that represent the Poincaré first-return map, also known as the Poincaré section.  Testing that a discrete map satisfies Liouville’s theorem is as simple as checking that the determinant of the Floquet matrix is equal to unity.  When the dynamics are represented in a Poincaré plane, these maps are called area-preserving maps.

There are many famous examples of area-preserving maps in the plane.  The Chirikov Standard Map is one of the best known and is often used to illustrate KAM theory.  It is a discrete representation of a kicked rotater, while a kicked harmonic oscillator leads to the Web Map.  The Henon Map was developed to explain the orbits of stars in galaxies.  The Lozi Map is a version of the Henon map that is more accessible analytically.  And the Cat Map was devised by Vladimir Arnold to illustrate what is today called Arnold Diffusion.  All of these maps display classic signatures of (low-dimensional) Hamiltonian chaos with periodic orbits hemming in regions of chaotic orbits.

Chirikov Standard Map
Kicked rotater
Web Map
Kicked harmonic oscillator
Henon Map
Stellar trajectories in galaxies
Lozi Map
Simplified Henon map
Cat MapArnold Diffusion

Table:  Common examples of area-preserving maps.

Lozi Map

My favorite area-preserving discrete map is the Lozi Map.  I first stumbled on this map at the very back of Steven Strogatz’ wonderful book on nonlinear dynamics [3].  It’s one of the last exercises of the last chapter.  The map is particularly simple, but it leads to rich dynamics, both regular and chaotic.  The map equations are

which is area-preserving when |B| = 1.  The constant C can be varied, but the choice C = 0.5 works nicely, and B = -1 produces a beautiful nested structure, as shown in the figure.

Iterated Lozi map for B = -1 and C = 0.5.  Each color is a distinct trajectory.  Many regular trajectories exist that corral regions of chaotic trajectories.  Trajectories become more chaotic farther away from the center.

Python Code for the Lozi Map

"""
Created on Wed May  2 16:17:27 2018
@author: nolte
"""
import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt

B = -1
C = 0.5

np.random.seed(2)
plt.figure(1)

for eloop in range(0,100):

    xlast = np.random.normal(0,1,1)
    ylast = np.random.normal(0,1,1)

    xnew = np.zeros(shape=(500,))
    ynew = np.zeros(shape=(500,))
    for loop in range(0,500):
        xnew[loop] = 1 + ylast - C*abs(xlast)
        ynew[loop] = B*xlast
        xlast = xnew[loop]
        ylast = ynew[loop]
        
    plt.plot(np.real(xnew),np.real(ynew),'o',ms=1)
    plt.xlim(xmin=-1.25,xmax=2)
    plt.ylim(ymin=-2,ymax=1.25)
        
plt.savefig('Lozi')

References:

[1] D. D. Nolte, “The Tangled Tale of Phase Space”, Chapter 6 in Galileo Unbound: A Path Across Life, the Universe and Everything (Oxford University Press, 2018)

[2] H. S. Dumas, The KAM Story: A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov-Arnold-Moser Theory (World Scientific, 2014)

[3] S. H. Strogatz, Nonlinear Dynamics and Chaos (WestView Press, 1994)

How to Weave a Tapestry from Hamiltonian Chaos

While virtually everyone recognizes the famous Lorenz “Butterfly”, the strange attractor  that is one of the central icons of chaos theory, in my opinion Hamiltonian chaos generates far more interesting patterns. This is because Hamiltonians conserve phase-space volume, stretching and folding small volumes of initial conditions as they evolve in time, until they span large sections of phase space. Hamiltonian chaos is usually displayed as multi-color Poincaré sections (also known as first-return maps) that are created when a set of single trajectories, each represented by a single color, pierce the Poincaré plane over and over again.

The archetype of all Hamiltonian systems is the harmonic oscillator.

MATLAB Handle Graphics

A Hamiltonian tapestry generated from the Web Map for K = 0.616 and q = 4.

Periodically-Kicked Hamiltonian

The classic Hamiltonian system, perhaps the archetype of all Hamiltonian systems, is the harmonic oscillator. The physics of the harmonic oscillator are taught in the most elementary courses, because every stable system in the world is approximated, to lowest order, as a harmonic oscillator. As the simplest dynamical system, one would think that it held no surprises. But surprisingly, it can create the most beautiful tapestries of color when pulsed periodically and mapped onto the Poincaré plane.

The Hamiltonian of the periodically kicked harmonic oscillator is converted into the Web Map, represented as an iterative mapping as

WebMap

There can be resonance between the sequence of kicks and the natural oscillator frequency such that α = 2π/q. At these resonances, intricate web patterns emerge. The Web Map produces a web of stochastic layers when plotted on an extended phase plane. The symmetry of the web is controlled by the integer q, and the stochastic layer width is controlled by the perturbation strength K.

MATLAB Handle Graphics

A tapestry for q = 6.

Web Map Python Program

Iterated maps are easy to implement in code.  Here is a simple Python code to generate maps of different types.  You can play with the coupling constant K and the periodicity q.  For small K, the tapestries are mostly regular.  But as the coupling K increases, stochastic layers emerge.  When q is a small even number, tapestries of regular symmetric are generated.  However, when q is an odd small integer, the tapestries turn into quasi-crystals.

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
“”
@author: nolte
“””

import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt
plt.close(‘all’)
phi = (1+np.sqrt(5))/2
K = 1-phi     # (0.618, 4) (0.618,5) (0.618,7) (1.2, 4)
q = 4         # 4, 5, 6, 7
alpha = 2*np.pi/q

np.random.seed(2)
plt.figure(1)
for eloop in range(0,1000):

xlast = 50*np.random.random()
ylast = 50*np.random.random()

xnew = np.zeros(shape=(300,))
ynew = np.zeros(shape=(300,))

for loop in range(0,300):

xnew[loop] = (xlast + K*np.sin(ylast))*np.cos(alpha) + ylast*np.sin(alpha)
ynew[loop] = -(xlast + K*np.sin(ylast))*np.sin(alpha) + ylast*np.cos(alpha)

xlast = xnew[loop]
ylast = ynew[loop]

plt.plot(np.real(xnew),np.real(ynew),’o’,ms=1)
plt.xlim(xmin=-60,xmax=60)
plt.ylim(ymin=-60,ymax=60)

plt.title(‘WebMap’)
plt.savefig(‘WebMap’)

References and Further Reading

D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time (Oxford, 2015)

G. M. Zaslavsky,  Hamiltonian chaos and fractional dynamics. (Oxford, 2005)