Abstract
In this work we consider an n-degrees of freedom Hamiltonian system near an elliptic equilibrium point. The system is transformed to normal form with respect to a given modulus and up to a given order. Using standard techniques developed in [7,8] a lower estimate of the domain of convergence of the transformation as well as an upper estimate of the remainder are obtained. Then the notion of effective stability is introduced: We construct an initial ball (in a suitable norm) such that, after a given big time interval, a point starting in the ball still belongs to a slightly bigger ball. In other words, we bound the effect of diffusion on the motion. An optimization of the size of the ball with respect to several free parameters is performed. Then a simple application is done to a 3 degrees of freedom third degree Hamiltonian and, finally, to the effective stability of the L4,5 points in the spatial Restricted Three Body Problem (RTBP), working out, explicitly, the Sun-Jupiter case.
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