Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Are there many examples of completely integrable geodesic flows (in the sense of Liouville), with say n integrals $f_1,\cdots, f_n$ such that everywhere, the differentials $(df_1,\cdots,df_n)$ are linearly independant ?

(recall that in the usual definition of completely integrable flows, one only requires that these are independent almost everywhere or in a dense open set).

Thanks !

share|cite|improve this question
A trivial example: geodesic flow on Euclidean $n$-space is straight-line motion, and the momenta $p_1, \dots, p_n$ are first integrals with obviously independent differentials. – Hans Lundmark Oct 22 '13 at 21:09
Yes of course you're right, but what about a geodesic flow on a more complicated Riemannian manifold ? – user102589 Oct 23 '13 at 8:25

I cannot comment; thus, I am posting this as an answer.

It is really strong to ask for global action-angle coordinates (see Duistermaat's article on that matter for a general discussion). However, there is a whole class of examples of nontrivial integrable geodesic flows on Lie groups: look for "Euler equations of finite-dimensional Lie groups" by Mischenko and Fomenko.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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