Explicit, Multi-Map Symplectic Integrator for Three -Body Classical Trajectory Studies in Hyperspherical Coordinates
Burkhardt, Paul
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https://hdl.handle.net/2142/84146
Description
Title
Explicit, Multi-Map Symplectic Integrator for Three -Body Classical Trajectory Studies in Hyperspherical Coordinates
Author(s)
Burkhardt, Paul
Issue Date
2004
Doctoral Committee Chair(s)
Belford, R. Linn
Department of Study
Chemistry
Discipline
Chemistry
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Chemistry, Physical
Language
eng
Abstract
Symplectic integrators are well known for preserving the phase space volume in Hamiltonian dynamics and are particularly suited for problems that require long integration times. There is a general operator splitting method for developing explicit symplectic integration algorithms to any arbitrary even order for separable Hamiltonians where the position and momentum coordinates are uncoupled. Explicit symplectic integrators for general Hamiltonians are more difficult to obtain, but can be developed by a composition of symplectic maps if the Hamiltonian can be split into exactly integrable parts. No general technique exists for splitting any Hamiltonian of general form. Many three body problems in classical mechanics can be effectively investigated in symmetrized, hyperspherical polar coordinates, but the Hamiltonian expressed in these coordinates is non-separable. In molecular dynamics, the hyperspherical coordinates facilitate the validation and visualization of potential energy surfaces and for quantum reactive scattering problems, the coordinates eliminate the need for adjusting the wavefunction between product and reactant channels. An explicit symplectic integrator for hyperspherical coordinates has not yet been devised. This dissertation presents an explicit, multi-map symmetrized composition method symplectic integrator for three-body Hamiltonians in symmetrized, hyperspherical polar coordinates, specifically for classical trajectory studies in the plane.
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