11
votes
2answers
410 views

Book on the Three body Problem

Hi all, I am looking for a good book about the famous (infamous perhaps?) three body problem - both theoretical and numerical hardless and accomplishments. can you help? Thanks
15
votes
10answers
2k views

Open problems in PDEs, dynamical systems, mathematical physics

(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.) I am an undergrad in math ...
1
vote
1answer
341 views

Partial linearization near a hyperbolic fixed point--Classical scattering

I am currently reading the famous article "Universal Properties of Maps on an Interval" by Collet, Eckmann and Lanford related to the Feigenbaum-Coullet-Tresser universality. I am in particular ...
1
vote
3answers
183 views

Applied examples of (non)uniformly hyperbolic and/or ergodic systems

I try to give reference to completely applied examples of (non)uniformly hyperbolic and/or ergodic systems. With completely applied I don't mean an irrational rotation on the torus but from other ...
12
votes
1answer
1k views

The Dedekind Eta Function in Physics

This interesting little fellow popped up in some operator algebra (Witt / Virasoro Lie algebra) I was exploring, so I've been curious about where else the Dedekind $\eta$-function makes a cameo ...
1
vote
0answers
117 views

Rigid-body in a central field: orbital and attitude motion

Question I would like to find a nice set of explicit coordinates for the family (parametrised by angular momentum) of reduced systems representing a rigid-body in a central field in which the orbital ...
3
votes
1answer
150 views

Avalanche Principle for higher dimensional unimodular matrices ?

Hello everyone, I have a quick question for people working on quasi-periodic Schrodinger operators, Lyapunov exponents for Schrodinger cocycles or in other fields that might make them aware of this ...
3
votes
2answers
385 views

How to deal with the singular reduction of the Hamiltonian n body problem?

I would like to consider the reduced Hamiltonian $n$ body problem, but am struggling with the angular momentum reduction seeing as the $SO(3)$ action is not free and the reduction is singular. ...
3
votes
3answers
1k views

Flow of a Hamiltonian vector field

Smooth vector fields are in a one-to-one relationship with flows $\Phi: D \subseteq M\times \mathbb{R} \rightarrow M$, $$X_m = {\frac{d}{d t}}_{t=0} \Phi(m, t),$$ and by the symplectic form also with ...
7
votes
1answer
419 views

Generalization of Rigid Body Motion to arbitrary (compact) Lie Groups

The classical dynamics of a rigid body in three dimensions may be described as the motion of a point on a configuration space given by the Lie group $SO(3)$, governed by Euler's equations for rigid ...
5
votes
0answers
247 views

Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked Definition: the Second-Hand Lion trace distance $D_k$ Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...
-3
votes
1answer
2k views

Quantum dynamics on varieties and Salmon Prizes

Concluding Progressive Remarks A new finding is Bates and Oeding's preprint "Toward a salmon conjecture" (arXiv:1009.6181), with its reference to the Salmon Prize. The Salmon Prize (photo of the ...
4
votes
2answers
309 views

mechanics: convergence to an equilibrium point

Hello, this is a math forum, I know, but my question is about classical mechanics. I am looking for a general (but simple proof) of the very intuitive idea physicists have about the following ...
2
votes
0answers
658 views

Bessel functions in wave propagation and scattering

Is there a way to scale Bessel J(n,.) (Bessel of first kind) and Bessel H(n,.) (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher vlaues of n) and small ...
0
votes
1answer
373 views

Simple system of ODEs with periodic coefficients

I am stuck with a little problem that I cannot solve mith the standard methods I learn at university. I have a system of coupled ODEs: $f'(t) = P \cos(k t + \Phi_1) g(t)$ $g'(t) = Q \cos(k t + ...
4
votes
3answers
577 views

Homogeneous linear differential equation system with simple periodical coefficient matrix

Hello, I encountered the following system of linear first-order differential equations: $y'(z)=A(z) y(z)$ where $y(z): R \rightarrow R^2$ and $A(z)=\begin{pmatrix} 0 & B Cos(\alpha z + \Phi_b) ...
0
votes
0answers
238 views

Is an immersed Kronecker join always a multilinear variety on a Hilbert space?

The question asked is: Is the implicitization of an arbitrary-rank immersed Kronecker join always a multilinear variety on a Hilbert space? This is related to another MathOverflow question ...
8
votes
0answers
355 views

Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks! Definition ...
5
votes
3answers
856 views

Bertrand theorem - central forces

Here is a version of Bertrand theorem. Let us consider a force $F(r)$ which depends only on the distance to a given point. If all trajectories which remain bounded are closed, then either $F(r)=ar$ ...
5
votes
3answers
522 views

Poincare Recurrence and Dense Sets

This is kind of a spin-off of the question asked here. Take the interval $X:=[0,1]$ with $\mu$ being standard Lebesgue measure. Let $f$ be a measure preserving map $f:[0,1]\rightarrow [0,1]$. The ...
12
votes
5answers
1k views

2- and 3-body problems when gravity is not inverse-square

Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing as $1/d^p$ for distance separation $d$ and some power $p$. Two questions: Presumably the 2-body ...
0
votes
0answers
222 views

What's a good approach to model this system?

Edited 15 Jul 2010 Willie's points are well-taken. I apologize for the wordy description. It turns out have a relative who is quite knowledgeable about numerical problems like this and has offered ...
8
votes
5answers
550 views

Persistence of fixed points under perturbation in dynamical systems

Suppose we have a smooth dynamical system on $R^n$ (defined by a system of ODEs). Assuming that the system has a finite set of fixed points, I am interested in knowing (or obtaining references about) ...
2
votes
1answer
634 views

Spectral curve of Elliptic Calogero-Moser systems

First, why all the coefficients in the characteristic polynomial of L are elliptic functions, since the diagonal entries of the matrix L are the momentums? second, how to understand the ramification ...
3
votes
0answers
178 views

What is known about first return times to Markov partitions for Anosov diffeomorphisms?

Consider an Anosov diffeomorphism $T: M \rightarrow M$ and a corresponding Markov partition $\mathcal{R}$ of $M$. For $x \in M$, let $\mathcal{R}(x)$ denote the element of $\mathcal{R}$ containing $x$ ...
64
votes
0answers
4k views

Dropping three bodies

Consider the usual three-body problem with Newtonian $1/r^2$ force between masses. Let the three masses start off at rest, and not collinear. Then they will become collinear a finite time ...
12
votes
2answers
827 views

Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?

That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian Ht on D2 such that the corresponding flow at time 1 is equal to $\psi$. I found this in claim a ...
58
votes
11answers
11k views

What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what is a non-integrable system.) In particular, is there a dichotomy between "integrable" ...
4
votes
4answers
413 views

Does it help to learn statistical mechanics in order to learn thermodynamic formalism?

Does it help to learn statistical mechanics or thermodynamics (as in physics or mathematical physics) in order to learn thermodynamic formalism: the study of equilibrium states, Gibbs measure, maximal ...