Three-Body Problem with Symmetry: Finite Masses and Spring Constants Considered

[Hans Stricker]

Consider the problem of three bodies two of which having mass M, one of them having mass m. Body m is in the middle between the other two, coupled to them by two equal linear springs in rest. Now fix the two bodies M and move body m for a small amount perpendicular to the connection line. Now let loose the three of them.

The solution is trivial when we assume m to be finite and M to be infinite (or vice versa), or when the spring constant is 0.

But the problem is so symmetric, that there might be hope to get a closed or approximate solution even for finite m >> M, M >> m, or even M = m?

 

[zhermes]

Definitely. Its a trivial problem in the lagrangian or hamiltonian formalism. If you are not familiar with those, think about conservation of momentum---and I think you can still get the right answer (and definitely the right idea of the answer).