Contact systems and contact integrators : 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
This thesis is concerned with the study of contact systems, which are ordinary differential equations whose flow preserves a contact structure. We study contact systems from both an analytical and numerical point of view. The traditional point of view is to study the Reeb vector field of a contact form. However, if the contact Hamiltonian vanishes then its contact vector field is not the Reeb vector field of any contact form equivalent to the given one. In this thesis we study exactly this case, when the contact Hamiltonian vanishes on some submanifold of phase space. This submanifold is invariant under the flow and we study the flow on it, including its stability and fixed points. The natural numerical method for a contact system is a 'contact integrator', a map that preserves the contact structure, which is suitable for exploring the long-time dynamics of contact systems. These have not been studied very much in geometric integration. In order to formulate our results and some consequences for contact integrators, we give a thorough development of the symplectification of a contact system and have found the integrable contact systems related to integrable homogeneous Hamiltonian systems via symplectification. We develop contact integrators by the splitting method, leading to an explicit contact integrator for any polynomial contact vector field. We also study how symplectic integrators for Hamiltonian systems and volume-preserving integrators for divergence-free systems are related to contact integrators for contact systems.