PERIODIC-SOLUTIONS TO SOME PROBLEMS OF N-BODY TYPE

We prove the existence of at least one T-periodic solution to a dynami cal system of the type [GRAPHICS] where the potentials V(ij) are T-per iodic in t and singular at the origin, u(i) is-an-element-of R(k), i = 1, ..., n, and k greater-than-or-equal-to 3. We also provide estimate s on the H-1 norm of this solution. The proofs are based on a variant of the Ljusternik-Schnirelman method. The results here generalize to t he n-body problem some results obtained by BAHRI & RABINOWITZ on the 3 -body problem in [6].