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Results tagged with sg.symplectic-geometry
Search options questions only
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user 12617
Hamiltonian systems, symplectic flows, classical integrable systems
1
vote
0
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304
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Lagrangian complement in a symplectic vector bundle
A standard, folk result in symplectic geometry states that:
in a symplectic vector bundle $(E,\pi,B,\omega)$, any lagrangian subbundle $L$ admits a lagrangian complement $L'$.
Having to use this …
- 4,306
5
votes
1
answer
586
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Is there a Legendrian Neighbourhood Theorem also for non-cooriented contact manifolds?
Context
According to Arnol'd, a contact structure on a smooth manifold $M$ is given by a corank 1 tangent distribution $C$ which is maximally non-integrable; this means that, for any local $1$-form …
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10
votes
1
answer
2k
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The universal property of the Liouville $1$-form
I am not totally sure if this question is appropriate for MathOverflow, or if it more adeguate to MathStackexchange.
As usual any feedback is welcome.
Introduction
Given an arbitrary smooth manifold $ …
- 4,306
3
votes
1
answer
1k
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What foliations are symplectic foliations?
On a manifold $M$, let $\mathcal F$ be a foliation having even-dimensional orientable leaves. I was wondering under what hypothesis I can state that $\mathcal F$ is the symplectic foliation of a Poiss …
- 4,306
5
votes
0
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382
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Sophus Lie on the symplectic foliation theorem
Given a Poisson manifold $(P,\{\cdot,\cdot\})$, its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e.
$$\mathcal C=\o …
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43
votes
2
answers
4k
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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In a widely circulated but unpublished letter in 1965, Palais explained the symplectic formulation of Hami …
- 4,306
6
votes
3
answers
957
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Given a vector field all of whose integral curves are closed, is the period a smooth function?
Disclaimer: The original question consisted of two parts. The first one
has been answered negatively (see
below the answers of Sam Lisi and
Alejandro). It remains the second one.
Background …
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18
votes
5
answers
2k
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Is there a coordinate-free proof of the hamiltonian character of the geodesic flow?
I do not know if this question is appropriate for this site, but I posted here without having answers, so now I make this attempt.
Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the tan …
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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 nonholonom...
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 transf …
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20
votes
4
answers
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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
10
votes
3
answers
1k
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How to motivate and interpret the geometric solutions of Hamilton-Jacobi equation?
Studying the Hamilton-Jacobi equation, I meet a generalization of the notion of its solutions, which is found already in the work of Sophus Lie.
For an H-J eqn, I mean a first order pde $H\circ dS …
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2
votes
1
answer
351
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On the Complete integrability of a tangent distribution
Reading about the geometrical theory of systems of first order pdes, I have met a result from symplectic geometry, that is easy to prove, but I am unable to give a reference for it. So my question is …
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3
votes
1
answer
618
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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 D …
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4
votes
2
answers
547
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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 …
3
votes
2
answers
593
views
What are the Killing vector fields on a triaxial ellipsoid?
Reading a paper on hamiltonian mechanics, in a section on classical examples of complete integrability, it is examined the geodesic flow of a triaxial ellipsoid.
Before separating the variables in th …
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