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Jesús Palacian

Reduction of resonant Hamiltonians with $N$ degrees of freedom


We deal with the analysis of Hamiltonian systems from a qualitative point of view, establishing the existence, stability and bifurcations of periodic solutions as well as the existence of some invariant tori. Our purpose is to illustrate the use of singular reduction on this problem. Reduction lowers the dimension of the problem under study; so, given that our first test problem is a two-degree-of-freedom Hamiltonian system in ${\mathbf R}^4$, it will be reduced to a Hamiltonian system of one degree of freedom on a two-dimensional surface called orbifold. The restricted three-body problem is considered as a benchmark for many years, and in particular there has been a bunch of works to obtain periodic solutions and related invariant sets around the equilibrium points $L_4$ and $L_5$. We shall illustrate how reduction theory is used in this context to establish the existence of these solutions rigorously.

Next we handle the $n$ degrees of freedom case where the polynomial invariants needed in the reduction theory are determined using an algorithm based on integer programming to obtain a Hilbert basis for a given resonance. The cardinality of the Hilbert basis is not known a priori but it is lower-bounded by $n^2$. After computing the Hilbert basis we use Gr\"obner bases and the division algorithm for multivariate polynomials to deal with the equations of motion in terms of the invariants. Besides we build the orbifold from the invariants and the constraints among them. Our aim is to reconstruct the periodic solutions of the full system from the critical points in the orbifold. To achieve it we use local symplectic coordinates around these points. We apply the theory getting some periodic solutions in some examples with $n = 3$.

This is a joint work with Ken R. Meyer and Patricia Yanguas.