Periodic Solutions of a Newtonian Equation: Stability by the Third Approximation

with z=q+ip, H=| 2T |z| +;1 4T |z| + } } } +;n&1 2nT |z| +r(t, z, z ). The coefficients |, ;1 , ..., ;n&1 are real and r is a T-periodic remainder of order o( |z| ), see [[1], appendice 7]. This is the Birkhoff Normal Form and the numbers ;1 , ..., ;n&1 ,... are the so called twist coefficients. They depend on the derivatives of f (evaluated at % ) up to the order 2n&1 and, when some of them is different from zero % is stable. This is a consequence of the Twist Theorem of Moser [16, 25]. In addition, the abstract theory of twist mappings can be applied to describe the dynamics of the equation in a neighborhood of % and to prove the existence of infinitely many subharmonic solutions [9, 18, 13]. article no. 0103