A system with a $2n$-dimensional phase space is Liouville-integrable if it admits $n$ independent first intgrals in involution.

Here integrable means that you can, in some way, solve the equations of motion by quadratures.

The Liouville-Arnol'd theorem states that a Liouville-integrable system admits a canonical transformation to action-angle coordinates, provided that it respects some other topological conditions.

These are that the level set of the first integrals must be compact and connected. My question is: is this condition very restrictive in the usual case? And does it imply that the orbit is quasi-periodic under those conditions?

I wonder if a problem like the Kepler problem with an open orbit (when the energy is greater than zero) is treatable with the Arnold-Liouville method.


1 Answer 1


Perhaps the paper with the self-explanatory title Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds could help.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.