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    Analysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand
    (Massey University, 2020) Offen, Christian
    Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis (a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived. (b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem. (c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for $A$-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations. (d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers' equation, KdV, fluid equations,...) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation. (e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed.
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    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
    (Massey University, 2016) Wilkins, Matthew Colin
    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.
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    The algebraic structure of B-series : a thesis presented in total fulfillment of the requirements for the degree of Master of Science in Mathematics at Massey University, Palmerston North, New Zealand
    (Massey University, 2010) Benn, James
    Runge-Kutta methods are some of the most widely used numerical integrators for approximat- ing the solution of an ordinary di erential equation (ODE). These methods form a subset of a larger class of numerical integrators called B-series methods. B-Series methods are expressed in terms of rooted trees, a type of combinatorial graph, which are related to the vector eld of the ODE that is to be solved. Therefore, the conditions for B-series methods to preserve important properties of the solution of an ODE, such as symplecticity and energy-preservation, may be ex- pressed in terms of rooted trees. Certain linear combinations of rooted trees give conditions for a B-series to be Energy-preserving while other linear combinations give conditions for a B-series to be Hamiltonian. B-series methods may be conjugate (by another B-series) to an Energy-preserving or an Hamiltonian B-series. Such B-series methods are called conjugate-to-Energy preserving and conjugate-to-Hamiltonian, respectivley. The conditions for a B-series to be conjugate-to-Energy preserving or conjugate-to-Hamiltonian may also be expressed in terms of rooted trees. The rooted trees form a vector space over the Real numbers. This thesis explores the algebraic structure of this vector space and its natural energy-preserving, Hamiltonian, conjugate-to-Energy preserving and conjugate-to-Hamitlonian subspaces and dual subspaces. The rst part of this thesis reviews important concepts of numerical integrators and introduces the general Runge-Kutta methods. B-series methods, along with rooted trees, are then introduced in the context of Runge-Kutta methods. The theory of rooted trees is developed and the conditions for a B-series to be Hamiltonian or have rst integral are given and discussed. In the nal chapter we interpret the conditions in the context of vector spaces and explore the algebraic structure of, and the relationships between, the natural vector subspaces and dual spaces.
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    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
    (Massey University, 2003) Joo, Seung-Hee
    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.