A solvable N-body problem in the plane. I

Abstract

We introduce and discuss an n-body problem in the plane, characterized by equations of motion of Newtonian type, r→̈j = Σnk=1F→jk, j = 1,..,n, with given "forces" F→jk having the following characteristics: F→jk depends only on r→j,r→k,r →̇j,r→̇k (i.e., only "one-body" and "two-body" forces are present); F→jk behaves as a (two-dimensional) vector under rotations in the plane (i.e., the model is "rotation-invariant"); for j = k,F→jk is linear in r→j and r→̇j; for j ≠ k, F→jk = |r→j - r→k|-2f→jk with f→jk a homogeneous polynomial of third degree in r→j,r→k,r →̇j,r→̇k (hence F→jk is homogeneous of degree one in r→j,r→k,r →̇j,r→̇k); F→jk contains linearly 8 arbitrary ("coupling") constants. The n-body problem is solvable for arbitrary n and for arbitrary values of the 8 coupling constants; its solutions display a rich phenomenology. If the 8 coupling constants are suitably restricted, the model is translation-invariant, and/or Hamiltonian; of course, when it is Hamiltonian, it is integrable; indeed in some case a Hamiltonian function can be explicitly displayed, as well as the corresponding Lax pair. © 1996 American Institute of Physics.