Spectral Methods for Hamiltonian Systems and Their Applications

Abstract

Hamiltonian systems typically arise as models of conservative physical systems and have many applications. Our main emphasis is using spectral methods to preserve both symplectic structure and energy up to machine error in long time. An engery error estimation is given for a type of Hamiltonian systems with polynomial nonlinear part, which is numerical verified by solving a Henon-Heiles systems. Three interesting applications are presented : the first one is the N-body problems. The second one is approximation for Weyl's Law and the third one is simulating quantum cooling in an optomechanical system to study the dissipative dynamics. Moreover, nonsmooth Hamiltonian systems problems are discussed for the limitation of this method which motivates our future work