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Galileo Galilei was the first modern scientist, launching a new scientific method that superseded, after one and a half millennia, Aristotle’s physics.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 ≈ t^{2})

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

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

1872 Jordan *Essay on the geometry of n-dimension*s

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

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*.

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

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

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

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

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

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)

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

1935 Bohr’s response to Einsteins “EPR” paradox

1935 Schrodinger’s cat

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

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

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 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