All Questions
11 questions
3
votes
0
answers
137
views
When the integrable system defines a moment map?
Assume $\mathcal{M}$ is a compact symplectic $2n$-dimensional manifold with a Hamiltonian action of the torus $\mathbb{T}^n$. Given a family of functions $F=(f_1,\ldots,f_n)$ defining an integrable ...
1
vote
0
answers
79
views
Liouville-Arnold and fibration relative to a convex polytope
Liouville-Arnold's theorem indicates that given a Hamiltonian torus action on a manifold and a set of $n$ functions $F$ from the manifold to $\mathbb{R}^n$ defining an integrable system, the pre image ...
1
vote
0
answers
44
views
When lagrangian fibrations are equivalent?
Given a $2n$-dimensional symplectic manifold $\mathcal{M}$ and two different lagrangian fibrations $\pi_1:\mathcal{M}\rightarrow \Gamma_1$ and $\pi_2:\mathcal{M}\rightarrow \Gamma_2$, with $\Gamma_1, \...
8
votes
1
answer
351
views
From time-dependent Hamiltonians to time-dependent symplectic/Poisson structures
Let $(M,\{.,.\})$ be a smooth Poisson manifold, and let $H\in C^\infty(M\times\mathbb{R},\mathbb{R})$.
Question: Does there exist $H_0\in C^\infty(M,\mathbb{R})$ and smooth parameter-dependent Poisson ...
- 1,959
2
votes
1
answer
376
views
algebraic momentum map
Let $T$ be a linear algebraic torus over $\mathbb C$ and $X$ be a smooth quasi-projective symplectic $T$-variety. Also, assume that the action of $T $ is free and $X/T$ exists as a smooth variety. Is ...
- 21
6
votes
0
answers
537
views
Hamiltonian dynamics on cotangent bundle
I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
- 341
2
votes
0
answers
129
views
Is there a notion of symplectic maps between spaces of volume forms on phase spaces?
For a $n$ dimensional smooth manifold $M$, I consider the cotangent bundle $T^*M$ with the canonical symplectic form $\omega$.
A symplectic map $\phi : T^*M \to T^* M$ is a map which leaves the ...
2
votes
0
answers
195
views
How to check conditions for Liouville-Arnold theorem? [closed]
Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem:
Let $F_1, \dots, F_n$ be $n$ functions in involution on a symplectic $2n$-...
- 679
10
votes
1
answer
2k
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.
...
- 243
4
votes
1
answer
396
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 ...
- 783
2
votes
1
answer
609
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 $(x_1,\...
- 243