A COMBINATORIAL PERTURBATION METHOD AND ARNOLD WHISKERED TORI IN VORTEX DYNAMICS

A combinatorial perturbation method for some n-body problems is presented. This method is used to reformulate a class of n-body problems in the form of near integrable Hamiltonian systems with many degrees of freedom. The KAM theory is worked out for n-body problems defined on subgraphs G of the complete graph K(n). The main result in this paper is the existence of Arnold's whiskered tori and diffusion for vortex lattice dynamics. Throughout the paper, combinatorial and graph-theoretic concepts play important roles, in particular the KAM nondegeneracy and Melnikov transversability conditions are given in terms of binary trees T(n). One of the main tools here is a combinatorial algorithm for symplectic transformations to Jacobi (relative) coordinates, which is based on binary trees T(n).