The Lie Transformation in Hamiltonian Dynamics: A perturbation approach to Hamiltonian Chaos
Prof. Yutaka Abe, Hokkaido Automotive Engineering College, Sapporo.
Abstract: The Lie transformation is a perturbation method for Hamiltonian systems. The idea of this method is to produce a near-identiy canonical transformation which simplifies the Hamiltonian. Lie transforms do not perturb the vector field itself, which consists of 2n scalar functions in a system of n degrees of freedom, but rather, perturb the Hamiltonian, a single scalar function. In the method of Lie transforms, we are concerned with generating a change of coordinates which will simplify the Hamiltonian. As examples, the undamped Duffing equation and Henon-Heeiles system are investigated to a certain extent with this method. In this note, we will reveal the characteristic features of this perturbation method and its limitation for Hamitonian chaotic dynamics.