"Symplectic Geometry Algorithms for Hamiltonian Systems" will be useful not only for numerical analysts, but also for those in theoretical physics, computational chemistry, celestial mechanics, etc. The book generalizes and develops the generating function and Hamilton-Jacobi equation theory from the perspective of the symplectic geometry and symplectic algebra. It will be a useful resource for engineers and scientists in the fields of quantum theory, astrophysics, atomic and molecular dynamics, climate prediction, oil exploration, etc. Therefore a systematic research and development of numerical methodology for Hamiltonian systems is well motivated. Were it successful, it would imply wide-ranging applications.
Preliminaries of Differential Manifolds.- Symplectic Algebra and Geometry Preliminaries.- Hamiltonian Mechanics and Symplectic Geometry.- Symplectic Difference Schemes for Hamiltonian Systems.- The Generating Function Method.- The Calculus of Generating Function and Formal Energy.- Symplectic Runge-Kutta Methods.- Composition Scheme.- Formal Power Series and B-Series.- Volume-Preserving Methods for Source-Free Systems.- Free Systems.- Contact Algorithms for Contact Dynamic Systems.- Poisson Bracket and Lie-Poisson Schemes.- KAM Theorem of Symplectic Algorithms.- Lee-Variational Integrator.- Structure Preserving Schemes for Birkhoff Systems.- Multisymplectic and Variational Integrators.