12

A system is completely integrable (in the Liouville sense) if there exist $n$ Poisson commuting first integrals. The Liouville-Arnold theorem, anyway, requires additional topological conditions to find a transformation which leads to action-angle coordinates and, in these set of variables, the Hamilton-Jacobi equation associated to the system is completely separable so that it is solvable by quadratures.

What I would like to understand is if the additional requirement of the Liouville-Arnold theorem (the existence of a compact level set of the first integrals in which the first integrals are mutually independent) means, in practice, that a problem with an unbounded orbit is not treatable with this technique (for example the Kepler problem with parabolic trajectory).

If so, what is there a general approach to systems that have $n$ first integrals but do not fulfill the other requirements of Arnold-Liouville theorem? Are they still integrable in some way?

ablagi
  • 125
  • 2
    Good question. You may find the following paper interesting, http://arxiv.org/abs/math/0210346. The statement is related to the fact that a Lie group with Abelian Lie algebra is a product of a torus and a vector space, but it's a bit more involved in that it's hard to control the behaviour of solutions of some equations at infinity. – Bedovlat Jul 27 '16 at 21:49
  • Crossposted from https://physics.stackexchange.com/q/267438/2451 – Qmechanic May 03 '17 at 14:15

1 Answers1

4

Let $M= \{ (p,q) \in \mathbb{R}^{n} \times \mathbb{R}^n \}$ ($p$ denotes the position variables and $q$ the corresponding momenta variables). Assume that $f_1, \cdots f_n$ are $n$ commuting first integrals then you get that $M_{z_1, \cdots, z_n} := \{ (p,q) \in M \; : \; f_1(p,q)=z_1, \cdots , f_n(p,q)=z_n \} $ with $z_i \in \mathbb{R}$ is a Lagrangian submanifold.

Observe that if the compactness and connectedness condition is satisfied then there exist action angle variables which means that the motion lies on an $n$-dimensional torus (which is a compact object).

The compactness condition is equivalent to that a position variable, $p_k$, or a momentum variable, $q_j$, cannot become unbounded for fixed $z_i$. Consequently, if the compactness condition is not satisfied there is no way you can expect to find action angle variables since action angle variable imply that the motion lies on a torus which is a compact object.

SillyMathematician
  • 178
  • Thank you for your time. In fact, I know the Liouville-Arnold theorem but, since I am interested in the physical application of it, I wonder if the compactness condition is "reasonable" or if there are many system that are not easily integrable for this reason. – ablagi Jul 13 '16 at 19:08
  • In the setting that the compactness condition is not there you could still try to apply certain reduction techniques to compactify your space. Then you can again return to the Arnold Liouville setting. The question about if the compactness condition is reasonable would physically amount to if it is reasonable that for finite energy their spatial or momentum variable can become unbounded. Does this answer your question? – SillyMathematician Jul 14 '16 at 13:03
  • OK, now it's clearer. So a motion with an open trajectory (like in the Kepler case in which there could be a parabolic or hyperbolic trajectory) is not treatable with this technique, right? Thank you. – ablagi Jul 14 '16 at 14:15
  • Not directly. You could compactify the original space. But then you are actually changing your underlying phase space in such a way that putting the first integrals constant you get a compact space. If you are not familiar with these techniques you could maybe post another question. Note I updated the answer based on the comments. – SillyMathematician Jul 14 '16 at 15:28
  • In the last paragraph shouldn't be the compactness equivalent to the bounded case and vice versa? If so, you have answered my question. Thank you – ablagi Jul 14 '16 at 15:45