A GENERALIZATION OF THE WEINSTEIN-MOSER THEOREMS ON PERIODIC-ORBITS OF A HAMILTONIAN SYSTEM NEAR AN EQUILIBRIUM

We study the Hamiltonian system (HS) (x)over dot = JH' (x) where H is an element of C-2(R-2N, R) satisfies H(0) = 0, H' (0) = 0 and the quad ratic form Q(x) = 1/2[H ''(0)x, x] is non-degenerate. We fix tau(0) = 0 and assume that R-2N congruent to E + F decomposes into linear subsp aces E and F which are invariant under the flow associated to the line arized system (LHS) (x)over dot = JH ''(0)x and such that each solutio n of (LHS) in E is tau(0)-periodic whereas no solution of (LHS) in F-0 is tau(0)-periodic. We write sigma(tau(0)) = sigma(Q)(tau(0)) for the signature of the quadratic form Q restricted to E. If sigma(tau(0)) n ot equal 0 then there exist periodic solutions of (HS) arbitrarily clo se to 0. More precisely we show, either there exists a sequence x(k) - -> 0 of tau(k)-periodic orbits on the energy level H-1(0) with tau(k) --> tau(0); or for each lambda close to 0 with lambda sigma(tau(0)) > 0 the energy level H-1(lambda) contains at least 1/2\sigma(tau(0))\ di stinct periodic orbits of (HS) near 0 with periods near tau(0). This g eneralizes a result of Weinstein and Moser who assumed Q\E to be posit ive definite.