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

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.