Friday, May 16, 2014
thought for the day/Celestial Mechanics and Mathematics
This is the famous 'Mandelbrot set'. It's a fractal that combines many 'Julia sets'(fractals) into one; it is of course infinit, and there's many videos of zooming in and out; any place leads to an infinity of other fractal forms including pictures of the whole Mandelbrot set that you start out with on the top picture.
"In our haste to press on with new applications and insights, let us not forget from where we came . . . we should reflect on the fact that ideas and properties that appear to have purely physical bases, such as stability and chaos itself, demand precise mathematical definitions if they are to be usefully applied." - Florin Diacu and Philip Holmes
Mathematicians have singled out the Navier-Stokes equation as one of the major unsolved mathematical problems of today. Just a few months ago, a Russian mathematician declared he had made a general proof; then, a little later, Terrence Tao pointed he had made some other progress; I can't claim to know what's true. It takes mathematicians two years or so to confirm some of the biggest new mathematical breakthroughs.
The Navier-Stokes equation is about the dynamics of uncompressible fluids. Solve it, and you solve perhaps a lot of materials science - not just chemistry, but the properties of materials composed of millions of atoms. The mathematics that has been thrown at it comes from Celestial mechanics which of course, that mathematics goes back to the Greeks Aristarchus at least. I'll be giving my notes from reading Diacu and Holmes "Celestial Encounters."
Isaac Newton's mechanics of course goes back to the motions of planetary bodies. The mathematics and science there of course goes back to Kepler, Galileo and Copernicus. Copernicus suggested a sun-centered universe; but, he still had perfect circles and Ptolemaic epicycles(which really goes back to Eudoxus, 4th century Greek mathematician which Plato partly admired and helped out and was a little jealous here and there as well!). The problems he had was not knowing the planets speed up and slow down at different points of their orbit(actually, I think if you read Koestler's "Sleepwalkers", I think he mentions that Copernicus did find something, but didn't know how seriously to take it . . . something very easy to do in at his time). Tycho Brahe of course built enormous sundials to track the movements and take the most accurate data to date. Kepler interpreted the data using the conics mathematics of Appollonius. This was all done before Galileo found his laws of falling bodies; that different weight bodies fall at the same rate in a vacuum. So, Kepler's mathematical achievement here is treamendous for the time! And of course, Isaac Newton came up with his inverse square law and actually derived from there the laws of Kepler; this is called the Kepler problem.
Here's a mechanical universe episode of the Kepler problem I couldn't make the embedding work, so . . .
Isaac Newton also started the idea of deriving what atoms an astronomical body is made of just by calculating its motions, or the motions of moons around it. After Isaac Newton, the mathematics of differential equations, calculus of variations of a hundred years since his time . . . from people like Leondard Euler, the Bernouli family, and of course Laplace and Lagrange . . . led to Laplaces's five volumes almost a thousand pages 'Celestial Mechanics.' It was outdated shortly by Hamilton(who innovated quaternions), and then Poincare(who systematized topology, came up with automorphic functions which combines elliptic functions with group theory and some other mathematics to solve celestial mechanics).
Mathematical physicists after Newton's time came up with conservation laws to calculate problems that involved more than two bodies. Newton also apparently looked for global solutions. This was a vectorial flow field. I've done the Kepler problem from a calculus book that is different than what's done in the mechanical universe video above. When checking out Newton's principia, I noted that the Kepler problem is solved in chapter one, and then he does a lot of his fluid dynamics and the not exactly rigorus treatment of determining the compositon of astronomical bodies by their bodies. I knew this, so I just read the first and last chapters and called it good. Now, I see I should probably check out some more. Also, I've learned that some of the other theorems proved in chapter one have other significance. First the theorem . . . lemma 28/chapter 1 has it that "No oval figure exists whose area, cut off by straight lines at will, can in general be found by means equations finite in the number of the terms and dimensions." This is one way of showing a transcendental number. The numbers pie and e were proven transcendental hundreds of years later in the 1800s. It of course comes out of celestial mechanics considerations. This just gives some more appreciation for Newton's 'Principia.' Some of the other major items of the Principia are the calculation of the precession of the equinoxes, the center point theorems. Newton found that to apply his mathematics to studying the motions of the planets, he needed to prove that all he had to take into account was the center point of the celestial body. He did the sphere, but then he also solved the oval shaped figure as well! There's also the classification of conics, and the use of a different curve for clockmaking than the conics. It's kind of parabolic but not quite a parabola.
Isaac Newton came up with much more mathematics. A twentieth century scholar wrote them all up in eight large thousand page volumes. I've only seen the first volume. It's this hugh book! It's like two feet long! His mathematics and Liebniz on the calculus opened up a floodgate of mathematics for the mathematicians of the 1700s. The main mathematicians have been mentioned already - the Bernoulli family(there was eight Bernoulli's in this mathematical family). They learned from Liebniz, they taught Leonard Euler. I have a pretty good video speech about Leonard Euler earlier in this blog posted. Then there was Lagrange, who did lots of good Number theory(so did Leonard Euler) after Fermat, and then Laplace. These and some others expanded the amount of what's called analyses. Analyses is the mathematician's name for anything calculus - differential equations, calculus of variations, complex analyses, and later, most in the 1800s, real analyses. Mathematicians from the birth of the calculus and analyses hoped to make closed form solutions to differential equations just like they did for algebraic equations(the famous quadratic formula, which goes back to the Babylonians of almost 2000 B.C.). There has been some headway in this direction in the work of Picard of the late 1800s and Sophus Lie. Part of the inspiration for Sophus Lie to make his lie algebras was to do exactly this. There's been work done even beyond Picard's first success. I'll get back to what's pointed out in "Celestial Encounters" now.
I think it was Poincare who came up with 'phase space.' I like to think of phase space as a kind of polar form. It's kind of like a Cartesian plane, but it it's a global picture of the energy/postion possibilities. Poincare I'm thinking showed that he could solve differential equations by a phase space diagram of the Newtonian vector flows.
Now we get into some more purely mathematical considerations. There's existence and uniqueness proofs. Mathematicians tried to prove these dynamical systems. And from a logical proof perspective, they'd make distinctions between existence and uniqueness solutions. They came up with examples of dynamical systems that violated either existence and others for uniqueness. If two curves meet at the same point, they violate uniqueness. To prove existence/uniqueness for these dynamical systems is what's called the navier-stokes equation problem.
In studying the existence/uniqueness proofs, subtle 'inferred units' emerge(see my Origin of mathematical knowledge article, third to first article of this blog). There's global and local existence, continuity. Continuity is generally defined 'locally.' When defined globally, it's called 'stability.'
Differential manifolds are a generalization of manifolds which comes from complex analyses. Manifolds, or topology comes out of complex analyses Jacob Bronowski 'inferred units' style. It's one of the biggest such mathematical events of recent mathematical history. Differential manifolds are roughly defined as when a surface can be locally approximated by a plance. Wildly different surfaces can then be considered equivalent locally. A torus and a sphere for instance. Differential manifolds reduce the amount of variables, or dimensions of a differential equation - hence helping to solve them. This process is related to conservation laws.
What Poincare was trying to solve that led to much of the above was the N-body problem. The N-body problem is the most general statement of the three body problem which Newton of course could not solve. It's asking for a global solution to the three and N-body problem. Differential calculus is more or less local, integrals are more or less global; Poincare found integral invariants in trying to solve the N-body problem. Poincare's exploration of the three and larger body problem led to what's called 'Chaos theory.'
James Glieck remarkes, I think it's more of a quote, in his "Chaos" book that quantum mechanics eats at Newtonian mechanics on the smallest scales. Einstein's theories of Relativity modifies Newton's mechanics at the largest scales. Chaos theory modifies Newtonian mechanics at the scales of the universe mathematical scientists thought was well established; the scale of the universe humans experience every day. Chaos happens in between stable 'Newtonian' periods of a dynamical system. On a phase space, a stable or equilibrium point is represented by a single point. Set the system in motion, and chaos potentially can happen, and then it dies down to an equilibrium point.
There's all kinds of inferred units that Poincare came up with in studying this chaos(which he called a strange attractor). Homoclinic points, and intersecting curves. There's a famous theorem of Poincare's; that there's two stable points on a ribbon where part of it is going in one direction and the other part is going in another direction. This led to the famous(amongst mathematicians and chaos theorists) Smale horseshow map. Smale imagine folding surfaces onto one another. This folding creates a horseshow pattern. Starting with two points, the two points chaotically drift from one another. They often get close together as well. Getting a little off topic for a moment. There's a prime distribution problem. Leonard Euler came up with these infinit series(really an equality of infinit series and infinity products) that somehow seems to calculate the distribution of primes; there's much more to say about this. I find that there's twin primes, and I've often felt that the distribution of primes is fractal and seems similar to what you see in this horseshow map(which is fractal). Getting back to Celestial mechanics and all . . . Smale made an equivalent expression of his horseshow map in terms of cantor dust.
Cantor dust goes back to George Cantor. George Cantor came up with transfinite numbers; or actually infinit sets of numbers. This goes back to Galileo actually. Galileo, in his "Two New Systems" found that one could make a mapping of the say even numbers to all the numbers; one could do likewise for the odd numbers to all the natural numbers. George Cantor found ways to do this for rational numbers, and all the negative integers. This was by means of a diagonal argument. George then found that he couldn't do this for real numbers. The set of real numbers are a larger infinity than the rational numbers, or the natural numbers. And, George found this process can be done to infinity and then you can start over and go to infinity again!
Cantor dust has to do with taking out thirds of a line segment, or even a third of every line segment unit off to infinity; If you sum the segments removed, they equal 1! Since the segment equals 1, the cantor dust equals 0! Going off on another tangent for a moment, this leads to measure theory and Lebesgue integrals, and 'real analyses.' They eventually are able to derive the fundamental theorem of the calculus from this new higher analyses, just like Newton derived the Kepler laws described above; and Newton's laws are derived from Einstein's General Relativity. Cantor dust is a fractal.
In this way, Smale relates the solution of differential equations to fractals. Things get far more technical; Smale relates his horeshow mapping to Poincare's homoclinic points(intersecting curves). This is kind of the celestial mechanics origin of Chaos theory. There is of course more.
A Kolmogorov studied the chaos theory of Hamiltonian systems. Hamiltonians are a generalization of Lagrange's dynamical systems theory, which is a reformulation of Newtonian mechanics in terms of calculus of variations. I'm certainly not explaining Hamiltonian equations. But, Kolmogorov studied the chaos theory of Hamiltonian systems. He cut out dynamical systems manifolds of quasi-periodic motion. This quasi-periodic motion looks like a torus. There's integrable and non-integrable hamiltons. Integrable hamiltonians lie on the torus. Non-integrable hamiltonians are when the tori are broken up.
There's a variety things I didn't fit in the above account of the celestial mechanics origin of chaos theory and relations to the Navier Stokes problem. Vladimir Arnold solved Hilbert's 13th problem which relates to KAM theory(the work of Kolmogorov), Euler found special phythagorean triangle solutions to the three body problem, Bifercations comes from Pontryagin, Liapunov exponents, Zhihong Xia solved a Painlev's conjecture about singularities. These singularities are about whether systems collide or not; they work out geometric diagrams that you would not believe for what sounds like something totally trivial.
A philosophical point brought up by Diacu and Holmes needs to be addressed. I've noticed this while reading James Glieck's "Chaos" as well. They both argue that chaos theory leads to a non-mathematical science; that mathematics is somehow the wrong approach. It certainly changes some of how scientists makes graphs of data, and say whether a theory is proved right or not based on those data graphs. But, my point is that chaos is about strange attractors, and strange attractors are abstractions just like the number two. Yes, we have a new abstraction. Chaos is a new science in the same way that quantum mechanics and General Relativity are new sciences. The goal becomes how does chaos theory relate to Newtonian mechanics just like Newtonian mechanics is derived from General relativity and Kepler's laws are derived from Newton's inverse square law. And the major problem of physics today is how are quantum mechanics and General relativity derived from one another? Which derives the other?
On the technology side, chaos theory had led to the technologies of going from stable to chaotic states; a strange attractor is a the whole of what could be many different stable patterns. Chaos theorists have been able to switch from one stable state to another of a given strange attractor! In any system whether electrical, mechanical, or chemical! This technology is generally not talked about as much as nanotechnologies and quantum computers; but, there is an already rather large literature and capability.
And then of course, if we solve the Navier Stokes equation, we get a materials science opened up as great as what nanotechnology can do for chemistry!