Symplectic integrators for vakonomic equations and for multi-Hamiltonian equations : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand

Abstract

Almost 200 years ago William Hamilton gave the world his reformulation of classical mechanics: the so-called Hamiltonian mechanics. By permitting a singular structure matrix, Mr Wilkins’ research extended this exalted theory to accommodate the Vakonomic equations, consequently allowing a solution to the sub-Riemannian geodesic and optimal control problems within this framework. The multi-Hamiltonian equation is an extension of Hamiltonian mechanics that appears in fields ranging from quantum mechanics to classical electrodynamics. Mr Wilkins’ research was conducted to the highest standards using numerical and theoretical proof and provided a stable, high-order multisymplectic numerical method for solving the multi-Hamiltonian equations where none previously existed. Our knowledge has increased because Hamiltonian mechanics has been extended to accommodate the Vakonomic equations and humanity now has a high-order multisymplectic numerical method for solving multi-Hamiltonian equations.