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  Interesting Hamiltonian System

+ 5 like - 0 dislike

The definition of a Hamiltonian system I am working with is a triple $(X,\omega, H)$ where $(X,\omega)$ is a symplectic manifold and $H\in C^\infty(X)$ is the Hamiltonian function.

I am wondering if someone can give me an interesting, or useful, example of a Hamiltonian system for which $X$ is not the cotangent bundle of a manifold.

This post imported from StackExchange Physics at 2014-08-07 08:04 (UCT), posted by SE-user JonHerman
asked Aug 6, 2014 in Theoretical Physics by JonHerman (25 points) [ no revision ]
Same question on Mathoverflow: mathoverflow.net/q/147395/13917 Related (since the two-torus cannot be a cotangent bundle): physics.stackexchange.com/q/126676/2451 and physics.stackexchange.com/q/32095/2451

This post imported from StackExchange Physics at 2014-08-07 08:04 (UCT), posted by SE-user Qmechanic
Also possibly of interest on MO: mathoverflow.net/questions/35900/…

This post imported from StackExchange Physics at 2014-08-07 08:04 (UCT), posted by SE-user Chris White

1 Answer

+ 3 like - 0 dislike

In many cases of interest, $X$ is a coadjoint orbit of a Lie group $G$, and $H$ an element in the corresponding Lie-Poisson algebra of the Lie algebra of $G$.  

These spaces describe in particular lots of exactly solvable problems - here $H$ is a sum of elements of the Lie algebra multiplied with Casimirs, plus a Casimir. Most nice exactly solvable problem can be cast in this form. See http://www.physicsoverflow.org/21556/coadjoint-orbits-in-physics?

For more on coadjoint orbits and their role in classical mechanics see 

J.E. Marsden and T.S. Ratiu,
Introduction to mechanics and symmetry,
Springer, New York 1994.


(A short notice is also in http://en.wikipedia.org/wiki/Coadjoint_representation )

answered Aug 7, 2014 by Arnold Neumaier (15,787 points) [ revision history ]
edited Aug 7, 2014 by Arnold Neumaier

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