# Interesting Hamiltonian System

+ 5 like - 0 dislike
3060 views

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
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

+ 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.

http://libgen.org/search.php?req=Marsden+symmetry

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

answered Aug 7, 2014 by (13,209 points)
edited Aug 7, 2014

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.