All Questions
9 questions
1
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0
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77
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Projecting phase space Liouville tori to configuration space in integrable systems
My question concerns whether or not the topology of a trajectory in the configuration space of an integrable system, or rather the area of configuration space accessible by trajectories can be ...
- 11
1
vote
0
answers
79
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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
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0
answers
44
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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, \...
6
votes
0
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536
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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
8
votes
1
answer
344
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Symplectic reduction of 4-manifolds with circle actions
Let $(M,\omega)$ be a $4$-dimensional closed symplectic manifold. Assume there exists a Hamiltonian $S^1$-action on $M$, let $\mu:M \to \mathbb{R}^*$ be its moment map and let $M_{\text{red}}=\mu^{-1}(...
- 843
2
votes
0
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192
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What is the relation between the different generating functions thought as finite approximations of action functionals
In the book Introduction to symplectic topology by MC Duff and Salamon, a discrete analogue of the action functional is defined on $\mathbb{R}^{2n}$. The idea is that a Hamiltonian isotopy can be ...
- 1,227
4
votes
1
answer
235
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Contradiction between fixed points of a hamiltonian diffeomorphism of a torus and quasi-periodic motion on a torus
Again a very simple question. I currently hold two contradictory ideas in my head
1) A hamiltonian diffeomorphism of a torus necessarily has fixed points
2) most hamiltonian actions on a torus in an ...
- 989
0
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0
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94
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On the measure of regular and chaotic regions in a phase space
Consider a Hamiltonian, non-linear, dynamical system associated to $H(\vec{q},\vec{p})$. Assume that the number of effective degrees of freedom is relatively small, say $D=3,4,5$. Now choose a certain ...
- 255
7
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0
answers
144
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Reference request: Liouville integrability of a torus action of small dimension on a symplectic manifold
Consider a hamiltonian toric acion on a connected real symplectic manifold of dimension 2n. The dimension of the torus, which we denote by $k$, may be less than $n$. The generators of the action will ...
- 4,787