2014/09/02 by Andrej Junginger, Junginger, Andrej, Jörg Main +3
Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Laser-Matter Interactions and Applications #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1409.0673
openalex publication_date 2014/09/02 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
Darboux's theorem guarantees the existence of local canonical coordinates on\nsymplectic manifolds under certain conditions. We demonstrate a general method\nto construct such Darboux coordinates in the vicinity of a fixed point of a\nnoncanonical Hamiltonian system via normal form expansions. The procedure\nserves as a tool to naturally extract canonical coordinates and at the same\ntime to transform the Hamiltonian into its Poincare-Birkhoff normal form. The\nmethod is general in the sense that it is applicable for arbitrary degrees of\nfreedom, in arbitrary orders of the local expansion, and it is independent of\nthe precise form of the Hamiltonian. The method presented allows for the\ngeneral and systematic investigation of noncanonical Hamiltonian systems in the\nvicinity of fixed points, which e.g. correspond to ground, excited or\ntransition states. As an exemplary field of application, we discuss a\nvariational approach to quantum systems which defines a noncanonical\nHamiltonian system for the variational parameters and which directly allows to\napply transition state theory to quantum mechanical wave packets.\n