All Questions
Tagged with ds.dynamical-systems sg.symplectic-geometry
64 questions
5
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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 ...
- 1,389
-3
votes
1
answer
2k
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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 ...
2
votes
0
answers
285
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In search for a more geometric proof of a result of van der Schaft and Maschke on nonholonomic mechanics
Edit: Now I have found something that appears to answer my own question. It is section 2 in the paper "On Submanifolds and Quotients of Poisson and Jacobi Manifolds" by Ch.-M. Marle. (There, he ...
- 4,306
2
votes
0
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356
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Dissipative Hamiltonian System with a Periodic Force
Let $H:P \to \mathbb{R}$ be a Hamiltonian on a symplectic manifold $(\omega,P)$ and let $X_H: P \to TP$ be the Hamiltonian vector-field. Let $F:P \to T^*P$ be a dissipative force field such that for $...
- 614
2
votes
1
answer
671
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How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups?
What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 ...
- 23
20
votes
4
answers
3k
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What is the role of contact geometry in the hamiltonian mechanics?
Let us assume someone is interested in the study of Hamiltonian mechanics.
What are good examples to illustrate him of the usefulness of contact geometry in this context?
On one hand the Hamiltonian ...
- 4,306
3
votes
1
answer
622
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On degenerate integrable hamiltonian systems
Is there some reference where the existence of local generalized action-angle variables is discussed in some detail for concrete examples of hamiltonian systems of mechanical type?
After Dazord and ...
- 4,306
4
votes
2
answers
550
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The fibers of the momentum map for the $SO(n+1)$ symmetry of the geodesic flow on $S^n$
My question is: Are the orbits of the geodesic flow on $S^n$ determined as the fibers of the momentum map for its $SO(n+1)$ symmetry?
I started by considering the analog problem for the orbits of the ...
0
votes
0
answers
261
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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
In ...
- 1,389
4
votes
2
answers
2k
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On the proof of the hamiltonian flow box theorem
The hamiltonian flow box theorem, as stated in Abraham and Marsden's Foundations of Mechanics, says that:
Given an hamiltonian system $(M,\omega,h)$ with $dh(x_0)\neq 0$ for some $x_0$ in $M$, there ...
- 4,306
24
votes
5
answers
2k
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When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riemannian manifold?
Here is my precise question. Let $M, \omega$ be a symplectic manifold and let $H: M \to \mathbb{R}$ be any smooth function. The symplectic form gives rise to an isomorphism between the tangent ...
- 29.2k
14
votes
1
answer
4k
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Floer homology and status of the Arnold conjecture
The Arnold conjecture on a closed symplectic manifold $(M,\omega)$ says in the weakest version that for a non-degenerate Hamiltonian there are at least $k$ 1-periodic orbits where $k$ is the sum of ...
- 2,590
16
votes
2
answers
2k
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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 ...
- 3,203
4
votes
3
answers
1k
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What are some conserved quantities of Poisson brackets?
Poisson brackets play the very important roles in Symplectic geometry and Dynamical system. I'm interested in some conserved quantities of Poisson brackets.
Let's say we are working on T^n x R^n (T^n ...
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