Perturbation theory and control in classical or quantum mechanics by an inversion formula

Abstract

We consider a perturbation of an "integrable" Hamiltonian and give an expression for the canonical or unitary transformation which "simplifies" this perturbed system. The problem is to invert a functional defined on the Lie-algebra of observables. We give a bound for the perturbation in order to solve this inversion. And apply this result to a particular case of the control theory, as a first example, and to the "quantum adiabatic transformation", as another example.

Enhancement of particle trapping in the wave-particle interaction

“…whose Hamiltonian can be written as H = H 0 + V , where H 0 is integrable and V a perturbation of order ǫ (compared to H 0 ). The results we use here have been proven rigorously [7,8]. In practice, it can be shown that a suitable control term f of order ǫ 2 exists such that H 0 + V + f has an invariant torus at a given frequency ω 0 .…”

Controlling chaos in wave-particle interactions

“…[27] for conservative systems that can be described by a near-integrable Hamiltonian of the form H = H 0 + εV , where H 0 is integrable and εV is a small perturbation with ε 1. For ε = 0, the dynamics is integrable, and the phase space presents just invariant tori.…”

“…[27] and presented in Refs. [28][29][30], which consists of the addition of a small perturbation to the Hamiltonian.…”

Hamiltonian control used to improve the beam stability in particle accelerator models

“…The goal of this paper is to present a reliable improvement of beam stability by increasing the DA in a simplified accelerator model, consisting of only one type of element having a sextupole nonlinearity [2,3,4,5]. We work in the framework of the Hamiltonian Control Theory presented in [7,8], where two methods of controlling symplectic maps have been described, namely using Lie transformations and generating functions. In the present paper we use the former method, which allows direct determination of the new controlled map; avoiding the possible problems related to coordinate inversion.…”