Two interacting bodies in an external field - MathOverflow

Two interacting bodies in an external field

2010-08-29T09:20:02Z

Hope, MO is the right place for this question (if not so: where would you pose it?).

Consider a two-body system in classical mechanics. As long as the interaction depends only on the distance of the two bodies, the two-body problem is integrable/solvable. Now consider the two bodies in a fixed external field. (This is only one step away from a three-body system that is known to be non-integrable in general, but obviously different from it.)

Question: Can the conditions on the combination of interaction and external field be explicitely given for the problem to be integrable/solvable?

It might be the case that the problem is always solvable. In this case the following reference request becomes predominant:

Reference request: Where can I find an explicit and elaborated treatment of this problem?

Answer by Richard Montgomery for Two interacting bodies in an external field

2011-09-25T03:00:04Z

I seriously doubt there is any general criteria. However there are more than one beautiful explicit examples of an external field which lead to an integrable problem. The simplest and probably best known is that of a constant field. An absolutely beautiful description of this and its solution can be found in the book `Essais sur le Mouvement des Corps Cosmiques' by V. Beletski, ch. 3. (See eq. 3.2.1) It illustrates the plethora of qualitatively different phenomenon possible within a single integrable system.

As a wierd tangent, the `anisotropic Kepler problem': keep the same potential but change the kinetic term to $a p_x ^2 + b p_y ^2$, $a \ne b$ is known to be non-integrable and Gutzwiller made an early career on this problem.