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34
votes
2answers
2k views

About a letter by Richard Palais of 1965.

In Cushman and Bates, Global Aspects of Classical Integrable Systems, 1997, I have read In a widely circulated but unpublished letter in 1965, Palais explained the symplectic formulation of ...
30
votes
5answers
1k views

are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?

I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...
9
votes
1answer
312 views

An algebraic Hamiltonian vector field with a finite number of periodic orbits

Edit: There is an interesting complete answer for the second part(see the answer by Thomas Kragh). I search for an answer for the first part. 1.Is there a polynomial Hamiltonian ...
6
votes
1answer
198 views

Generalizing “variation of parameters”

I'm stuck on generalizing an ODE formula and could use your help! One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here ...
6
votes
1answer
305 views

Lifting a Diffeomorphism to the Cotangent Bundle

Both Abraham-Marsden and Da Silva seem to imply that given a symplectomorphism $g:T^\ast X\to T^\ast X$ which preserves the tautological $1$-form $\alpha$, it must be that $g$ is fibre preserving. ...
6
votes
1answer
315 views

Calogero-Moser system: relationship between dual variables and the KKS construction

This is a question about the relationship between two ways of viewing the Calogero-Moser system. $$\ddot x_i=2\sum_{j\neq i}\frac{1}{(x_i-x_j)^3}\qquad i=1,\ldots N$$ By introducing the $N$ ...
3
votes
2answers
169 views

Momentum a cotangent vector

Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities. Furthermore, many sources ...
3
votes
1answer
159 views

Weinstein's local classification of Lagrangian foliations

In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle. I ...
3
votes
0answers
246 views

Why is geometric quantization (esp. Berezin-Toeplitz quantization) interesting for a symplectic geometer/topologist?

I know that many symplectic geometers are interested in quantization as well. From what I understood, quantization isn't expected to be used as a tool to answer symplectic questions (as in ...
2
votes
0answers
87 views

Is it true that a nondegenerate minimizing periodic orbit of mechanical Hamiltonian system is hyperbolic

Consider mechanical Hamiltonian system of the form $$H(p,q)=\dfrac{\Vert p\Vert^2}{2}+V(q),\quad (q,p)\in T^*\mathbb T^n.$$ Here we suppose the periodic orbit $\gamma$ minimizes the Lagrangian ...
1
vote
1answer
212 views

Computation with the Legendre Transform

Let $M$ be a manifold and fix a Lagrangian $L\in C^\infty(T M )$. Let $x_1,\dots x_n$ be local coordinates for $M$ and equip the tangent bundle and cotangent bundle with standard coordinates ...
1
vote
0answers
47 views

Id monodromy in hamiltonian dynamics

In my problem I have non autonomous Hamiltonian which depends on 2 parameters (pretty close to oscillator Hamiltonian, $(a+b\cos t +1) p^2+(a+b\cos t-1)q^2$, $a,b$ - parameters). From numerical ...
0
votes
0answers
192 views

How to understand the matrix behind a Hamiltonian?

thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which resembles a second quantized model taking the particles to be ...