A Lie-Algebra version of Classical or Quantum Hamiltonian Perturbation Theory and Control, with examples in Plasma Physics

Abstract:

How to describe the Perturbation Theory of a non-integrable Hamiltonian? Which property do we want to preserve after perturbation? We propose to use a sub-Lie Algebra of the space of "Observables".

So we consider an Hamiltonian in a sub-Lie Algebra B (which we name "admissible") of the Lie-Algebra of "Observables".

For a "quite general but small" perturbation of this Hamiltonian, we give an expression for the sub-Lie-algebra isomorphic to B which contains the perturbed system. More precisely we give an expression for the automorphism ("change of variables") which conjugates the 2 sub-Lie-algebras.

A simpler problem is to "slightly" modify the perturbed system (by an additive "control" term, for instance quadratic in the perturbation) such that the above automorphism is simple to compute.

This theory generalizes a recent control of Hamiltonian systems that has been already applied in some physical examples. Here we give some other examples, mainly in Plasma Physics.