Construction of Darboux coordinates in noncanonical Hamiltonian systems via normal form expansions

TL;DR

This paper presents a general method to construct Darboux coordinates in noncanonical Hamiltonian systems near fixed points using normal form expansions, facilitating analysis of complex dynamical systems.

Contribution

The authors develop a systematic, order-independent approach to derive Darboux coordinates in noncanonical Hamiltonian systems applicable to arbitrary degrees of freedom.

Findings

Method effectively transforms Hamiltonian into Poincare-Birkhoff normal form.

Applicable to systems with arbitrary degrees of freedom and expansion orders.

Enables systematic study of noncanonical systems near fixed points.

Abstract

Darboux's theorem guarantees the existence of local canonical coordinates on symplectic manifolds under certain conditions. We demonstrate a general method to construct such Darboux coordinates in the vicinity of a fixed point of a noncanonical Hamiltonian system via normal form expansions. The procedure serves as a tool to naturally extract canonical coordinates and at the same time to transform the Hamiltonian into its Poincare-Birkhoff normal form. The method is general in the sense that it is applicable for arbitrary degrees of freedom, in arbitrary orders of the local expansion, and it is independent of the precise form of the Hamiltonian. The method presented allows for the general and systematic investigation of noncanonical Hamiltonian systems in the vicinity of fixed points, which e.g. correspond to ground, excited or transition states. As an exemplary field of application, we discuss a variational approach to quantum systems which defines a noncanonical Hamiltonian system for the variational parameters and which directly allows to apply transition state theory to quantum mechanical wave packets.