(Part I of Colloquium Talk) - zung.zetamu.netzung.zetamu.net/Files/2018/03/colloquium-postech-2018.pdfNguyen Tien Zung (IMT) Colloquium Postech 2018 March 16th 2017 20 / 35. The world

This amazingly periodic world(Part I of Colloquium Talk)

Nguyen Tien Zung

Institut de Mathematiques de Toulouse, Universite Paul SabatierVisitor at Center for Geometry and Physics, IBS

Postech, March 16th 2018

Main ideas of this talk

Things that we can observe in the world are very often of periodicnature. In mathematics, this phenomenon corresponds to the notionof integrability (a la Liouville) of dynamical systems.

Starting from late 19th, early 20th century: most abstract dynamicalsystems are shown to be non-integrable (e.g., the 3-body problem, byPoincare), chaotic (unpredictable), ergodic (mixing).

Apparent paradox: why most systems are non-integrable, chaotic, butthe world is still periodic?

Explanations involving physics, mathematics, and philosophy: physicallaws, mathematical theory of integrable systems and theirperturbations, decomposition of dynamical systems and the classical5-element philosophy.

Nguyen Tien Zung (IMT) Colloquium Postech 2018 March 16th 2017 2 / 35

Plan of the talk

1 The world is periodic

2 Mathematical theories behind periodic motions

3 The world is chaotic

4 Revisiting the 5-element theory (ohaeng)

5 Why the world is so periodic anyway?


Eppur si muove: Galilei (1564-1642)

The Earth moves around the Sun in a periodic way (period = 1 year =365.24219858156 days, calculated by Omar Khayyam in XII century)


Other examples of periodicity

The Moon turns around the Earth (period = 27 days, but do youknow why a Lunar moth has 30 days but not 27 days?)

Jupiter moves around the Sun (period ≈ 12 years, corresponding tothe 12 animals in the ”Chinese zodiac”?)

Sound waves: The musical note A4 (la) has frequency 440Hz byinternational standards, i.e. its period is 1/440 s.

Light waves (visible light has frequency range from 430THz to 777THz), water waves, etc.

The motors, the wheels, the watches, etc.

The internal motion of molecules

Our everyday life: sleep - work - food - fun - sleep

Cycles in the society (economic, political, etc.)

Reincarnation theory of Buddhism


Night and day

The Earth moves around itself (period = 24 hours), creating night and day.


Why is the world so periodic?

- Simplistic answer: God created it this way- Deeper answers: need mathematics and physics

Sound waves created by periodic pressure/depressure of the air.


From Ptolemy to Liouville

Claudius Ptolemy (c. 170 - c. 170, in Egypt during Roman Empire) is oneof the most famous mathematicians and astronomers of all times. Hisbooks were reference books during a thousand years. Created an epicyclemodel of planetary motions, based on the ideas on Hipparchus (190-120BC) to explain the apparent ”retrograde motion” of the planets on the sky.

Mars retrograde.


From Ptolemy to Liouville

Ptolemy and his epicycle model of planetary motions: physically wrong

(geocentric, not heliocentric), but mathematically very good!


The tori

Periodic motion = the orbit is a circle (an ellipse is also a circle,topologically). A circle is an 1-dimensional torus T1 = R/ZQuasi-periodic motion = has many components, each component isperiodic with its own period. Lives on a torus of dimension n where nis the number of components (the number of degrees of freedom)

Tn = (R/Z)n

A 2-dimensional torus (looks like a doughnut), whichi appears in Ptolemy’s model.


Kepler’s second law

Johannes Kepler (1571-1630), based on very exact observations of histeacher Tycho Brahe (1546-1601).- Conservation of angular momentum- First integral (function invariant by the motion) for a dynamical systemon the phase space.


Analytical celestial mechanics

Differential calculus: Newton Newton (1643-1727), Leibniz(1646-1716)

Newton’s law of universal attraction, leads to 2nd order equations inphysics. So the number of variables to describe the state of a systemis twice the number of variables to describe its position/configuration.(The initial condition includes not only the position, but also thevelocity). If the system has n degrees of freedom then its phase spacehas dimension 2n.

Euler (1707-1783), Lagrange (1736-1813), Laplace (1749-1827),Poisson (1781-1840), Hamilton (1805-1865), etc.

They could write down the equations of motion, compute and predictthe positions of the planets, using analytical methods.

They could even find new planets mathematically before being able toobserve them in the sky (Le Verrier, planet Neptune in 1846).


Liouville’s Theorem

Joseph Liouville(1809-1882) was the first one to prove thequasi-periodicity of the systems mathematically

Integrabilitya la Liouville: Existence of n conservation laws ininvolution for a system with n degrees of freedom.(”In involution” means that their Poisson bracketswith respect to the symplectic form vanish).

Theorem (Liouville 1855)

If a system with n degrees of freedom is integrable a la Liouville then itsmotion (with a given initial condition) is quasi-periodic on a n-dimensionaltorus.

http://sites.mathdoc.fr/JMPA/PDF/JMPA_1855_1_20_A11_0.pdf


http://sites.mathdoc.fr/JMPA/PDF/JMPA_1855_1_20_A11_0.pdf

Approximation by trigonometric functions

Theorem (Hipparchus-Ptolemy-Fourier)

Quasi-periodic functions can be uniformly approximated, with arbitraryprecision, by trigonometric functions:

f (t) =∑i

ai sin(2πωi t + bi )

Hipparchus (190-120 BC): Greek mathematician and astronomer, founderof trigonometry. Joseph Fourier (1768-1830): French mathematician,famous for Fourier series.

Together with Liouville’s theorem: Long-term prediction of the state of anintegrable system (e.g., the positions of the planets) by usingtrigonometric functions.


A bit more about Liouville

Known for this theorems in pure mathematics: holomorphic functions,transcendental numbers.

Founder of Journal de Mathematiques Pures et Appliquees, one of theoldest and most famous mathematical journals in the world, stillexists today.

Liouville is the person who brought the work of Evariste Galois to theattention of the world, many years after Galois’s death.

Also a great applied mathematician and physicist.

Liouville’s theorem about periodic motion: probably one of the mostfundamental laws in physics.


The angles and the actions

- Spinning tops (children’s toys, gyroscopes, the Earth, ...)

Lagrange top.

- Three angles: rotation, precession, nutation. (3 = the dimension of theconfiguration space SO(3) = the number of degrees of freedom).- For the Earth: rotation period = 1 day, precession period = 26 thousandyears (the fixed stars shifted on the sky 1 zodiac sign after 2000 years,leading to conflict between Indian and Western astrological styles),nutation period = 18.6 years with a very small angle (17” in longitude)


Additional variables

Recall that if the configuration space N has n dimensions then phasespace = T ∗N has 2n dimensions (symplectic manifold)

In order to describe the state of a system, need not only the positionbut also the velocities (or momenta)

Correct equations in physics (and even in finance) must be of secondorder, and not first oder, in the position coordinates.

So besides the n angle variables, need n more variables.

The most important ones are called action variables. These actionvariables are constant on each Liouville torus (once we already fixedan initial condition, or fix the values of conserved quantities, e.g.energy, angular momentum), but they are functions on the wholephase space.


Action-angle variables

- Proved by Henri Mineur (French astrophysicist) in 1935. Reproved byV.I. Arnold and other people in 1960s. Now known asArnold-Liouville-Mineur theorem.- The symplectic form is canonical in action-angle variables:

ω =∑i

dpi ∧ dθi

https://en.wikipedia.org/wiki/Henri_Mineur

- Used by physicists, even before the proof of existence: Einstein 1917,quantization a la Bohr–Sommerfeld–Epsteinhttp://hermes.ffn.ub.es/luisnavarro/publicaciones_archivos/

Einstein’s_rule_(2000).pdf

- When the system is quantized: angles variables become phases of thewave functions (no way to measure them even though we know theyexist), and action variables become quantum numbers (measurable byinstruments)


https://en.wikipedia.org/wiki/Henri_Mineur

http://hermes.ffn.ub.es/luisnavarro/publicaciones_archivos/Einstein's_rule_(2000).pdf

http://hermes.ffn.ub.es/luisnavarro/publicaciones_archivos/Einstein's_rule_(2000).pdf

Perturbation and stability theory

- There are always small (or not so small) perturbations by external forces- Integrable systems are very robust/stable under perturbations- Kolmogorov–Arnold–Moser (KAM) theory, starting from 1950s: mostintegrable motions are still quasi-periodic under small perturbations.Andrei Kolmogorov (1903-1987), Vladimir Arnold (1937-2010), JurgenMoser (1928-1999): three famous 20th century mathematicians.- Nekhoroshev theory (from 1980s): Even if the motion is no longerquasi-periodic, it still looks like quasi-periodic for a very very long time.- What does it mean for us: Even though astronomers like Jacques Laskarhttps://en.wikipedia.org/wiki/Jacques_Laskar showed that theSolar system has many noises and will eventually be destructed one day(before the complete burn-out of the Sun), but its quasi-periodic behavior,as we know it, will last for at least a billion years.


https://en.wikipedia.org/wiki/Jacques_Laskar

A new kind of conservation laws

- The first proof of action-angle variables theorem are rather complicated,difficult to follow and to remember- New conceptual proof, based on a new kind of conservation laws:

Theorem

Anything which is preserved by a dynamical system is also preserved by itsassociated torus actions.

(If the system is integrable a la Liouville then the associated torus actionsare nothing else but Liouville torus actions)Immediate consequence: Liouville torus actions are Hamiltonian, and sowe get action functions (the components of the momentum map of thisHamiltonian action)This law has important consequences for other kinds of dynamical systemsas well, including non-holonomic systems, non-integrable systems.


The world becomes chaotic

Until late 19th century: people studied integrable systems and theirperturbations, and didn’t know about the existence of non-integrablesystems. The classical world was periodic.

Henri Poincare (1854-1912, ”the last universalist”) studied thegeneral 3-body problem, and found that this system is non-integrable!Explicit expressions for long-term solutions don’t exist, can’t predictthe future. Suddenly, the world is no longer periodic!

In 20th century: ”chaotic” or ”turbulent” systems become”fashionable”. Even very simple systems in low dimensions are shownto contain chaotic behavior


Simple examples of chaotic systems

The double pendulum (2 degrees of freedom) and Lorenz butterfly (ODE in R3)

Even though these systems are deterministic, it’s impossible to predict thebehavior of a solution over the long term, due to hyperbolicity (verysensitive on initial data, impossible to make long-term predictions).Butterfly effect.


Smale’s horseshoe

A very simple model of a hyperbolic (hence chaotic) deterministic system,has a region equivalent to the Bernoulli shift {0, 1}Z → {0, 1}Z (at eachstep, choose randomly between 0 and 1).

Smale’s horseshoe, and an invariant Cantor set.


A measure of chaos: entropy

What is chaos? Loss of order? Quick loss of information over thetime.The amount of information can be measured by entropy formula(Shannon, similar to Boltzmann’s entropy in thermodynamics): IfP = {P1, . . . ,Pn} is a partition of the whole space, and µ is theprobability measure, then

h(P) =∑i

µ(Pi )(− lnµ(Pi ))

Kolmogorov-Sinai entropy : measure of rate of loss of information of adynamical system

h = limn→∞

h(Pn)

nwhere Pn is the partition at time n created from a partition P at time0 by the flow (by making intersections of all the sets obtained by theflow at the times 0, 1. . . . , n).Quasi-periodic then entropy = 0, chaotic then entropy > 0.


Searching for integrable systems

Dynamical systems may depend on parameters (e.g. masses, type ofpotential, etc.)

Searching for integrable cases in the the ocean of all possibleparameters is like searching for a needle in a haystack. (Very few andfar in between)

Sofia Kovalevskaya (1850-1891, the first woman with a PhD inMathematics) found one famous integrable system bearing her name,by a method of analysis of singular points (Lyapunov-Kovalevskayaexponents): Kovalevskaya top.

A video of the simuation of this spinning top:https://av.tib.eu/media/10361


https://av.tib.eu/media/10361

Kovalevskaya Top

A model of Kovalevskaya’s top, and one of its orbits (2-dimensional quasi-periodic

orbit, but looks rather complicated)


Modern-day test of (non) integrability

Obstructions to integrability (topological, geometrical, algebraic,etc.). For example, Valery Kozlov (vice president of Russian Academyof Sciences): If the configuration space is 2-dimensional and have atleast 2 handles, then the system can’t be integrable.Modern tool: Differential Galois theory, similar to Galois theory foralgebraic extensions, but together with a differential operator, forshowing”non-solvability” or non-integrability of dynamical systems.

Theorem (Morales-Ramis-Simo, extended by Ayoul-Zung)

If a dynamical system is integrable then all of its differential Galois groupsare virtually Abelian.

J-P Ramis: French academician, my colleague in Toulouse. J. Morales andC. Simo: Spanish mathematicians. M. Ayoul: my student. With the abovetheorem, one detect the parameters at which the system might beintegrable, and then search for first integrals, to find really integrablesystems (or integrable potentials)


Theory vs reality: what goes wrong?

Theory since the time of Poincare: The world is mainlynon-integrable, chaotic. Integrable systems are very few and farbetween in the universe of dynamical systems.

Reality : Most of the things that we see around are still integrable,quasi-periodic. Even the chaotic things like the stock markets stillshow signs of periodicity (boom-bust cycles).

Why this discrepancy between theory and reality? What’s wrongthere? Ideas from physcis, mathematics, and philosophy:

Physics: Things need energy to move, and the energy function leadsto 2nd-order equations and oscillators (periodic motions). By the way,Black-Scholes model in finance is dead wrong because it’s of 1st orderand fails top model the energy of the market.

Mathematics: Mathematical theories of dynamical systems, inparticular about their long-term stability or instabilities.

Philosophy: revisit the 5-element theory.


5-element theory (ohaeng) revisited

EveryAsian person ”knows” thisclassical philosophical theory.Is it science or superstition?According toMarxism, this theory is rubbish,because, as we know, the worldis composed of more than 100elements (atoms), not just 5.But if it were rubbish,maybe it would have dieda long time ago. But it stillsurvives, so is there anythingto it? What is it really about?


Dynamical content of 5-element theory

The 5-element theory is really a theory about the decomposition ofdynamical systems into basic states and the interaction among these basicstates. Elements are basic states, not basic chemical elements.

metal = regular, recurrent

water = flowing, transport

wood = growing, expansion

fire = hyperbolic, chaotic

earth = contracting, getting solid

Closely related to modern theory of dynamical systems (various kinds ofdecomposition, e.g. spectral decomposition). In particular, Morse-typetheory of Charles Conley (1933-1984) for dynamical systems.

Theorem (Conley 1978, fundamental theorem of dynamical system)

Any dynamical system on a (compact metric) space M can bedecomposed into 2 parts: chain-recurrent and gradient-like.


Conley decomposition

Chain-recurrent and gradient-like elements.

Conley’s theorem has only 2 ”elements”, where are the other three?The key here is that the system is closed (not getting in our out of M). Ifnot closed then we find the other 3 elements, so that there are 5 in total:

1 going around inside (chain-recurrent)

2 going in the increasing direction of a function inside (gradient like)

3 getting in

4 getting out

5 going through


Simple model: Affine transformations of the plane

Discrete-time dynamical systems, linearized, generated by affinetransformations of the plane:

T (x) = A.x + b

where x is a small region near the origin of the plane, A is a square 2× 2matrix, b is a 2-dimensional vector.

1 b 6= 0: water (transport by b); otherwise b = 0:

2 Both eigenvalues have norm greater than 1: wood (expansion)

3 Both eigenvalues have norm smaller than 1: earth (contraction)

4 One greater than 1, one smaller than 1: fire (hyperbolic)

5 Both eigenvalues are complex of norm 1: metal (turning around)

Can also explain the generating cycle and the controlling cycle usingdynamical systems.


Why is the world so periodic anyway?

Some possible reasons:

Periodic systems are very very stable. There are many chaoticsystems but they disappear after a short time, while there are fewerperiodic systems but they are there to stay.

The law of natural selection applies not only to living creatures, butto all things. Things that last are things that can persist, and thestructures which are periodic are the most persistent.

Maybe there are many worlds, e.g. parallel worlds, and we are luckyto live in a regular world where living things can appear. Otherwisewe would not exist to wonder about the periodicity of the world.

The 5-element theory, and the theory of composition of dynamicalsystems, apply to everything (and not only to us), and the recurrenceplays a fundamental role in these theories. There is change in theperiodic things, and there is periodicity in the changing things.


Extraterrestrial Intelligence?

Many Earth-like planets in the Universe. Maybe on Kepler-186f (500 light years

from us) there are also living creatures thinking about the periodicity of the world?


THANK YOU FOR YOUR ATTENTION!


Documents

(Part I of Colloquium Talk) - zung.zetamu.netzung.zetamu.net/Files/2018/03/colloquium-postech-2018.pdfNguyen Tien Zung (IMT) Colloquium Postech 2018 March 16th 2017 20 / 35. The world