86 research outputs found
Three approaches towards Floer homology of cotangent bundles
Consider the cotangent bundle of a closed Riemannian manifold and an almost
complex structure close to the one induced by the Riemannian metric. For
Hamiltonians which grow for instance quadratically in the fibers outside of a
compact set, one can define Floer homology and show that it is naturally
isomorphic to singular homology of the free loop space. We review the three
isomorphisms constructed by Viterbo (1996), Salamon-Weber (2003) and
Abbondandolo-Schwarz (2004).
The theory is illustrated by calculating Morse and Floer homology in case of
the euclidean n-torus. Applications include existence of noncontractible
periodic orbits of compactly supported Hamiltonians on open unit disc cotangent
bundles which are sufficiently large over the zero section.Comment: 30 pages, 6 figures. To appear in J. Symplectic Geom. (Stare Jablonki
conference issue
Optimization on manifolds: A symplectic approach
There has been great interest in using tools from dynamical systems and
numerical analysis of differential equations to understand and construct new
optimization methods. In particular, recently a new paradigm has emerged that
applies ideas from mechanics and geometric integration to obtain accelerated
optimization methods on Euclidean spaces. This has important consequences given
that accelerated methods are the workhorses behind many machine learning
applications. In this paper we build upon these advances and propose a
framework for dissipative and constrained Hamiltonian systems that is suitable
for solving optimization problems on arbitrary smooth manifolds. Importantly,
this allows us to leverage the well-established theory of symplectic
integration to derive "rate-matching" dissipative integrators. This brings a
new perspective to optimization on manifolds whereby convergence guarantees
follow by construction from classical arguments in symplectic geometry and
backward error analysis. Moreover, we construct two dissipative generalizations
of leapfrog that are straightforward to implement: one for Lie groups and
homogeneous spaces, that relies on the tractable geodesic flow or a retraction
thereof, and the other for constrained submanifolds that is based on a
dissipative generalization of the famous RATTLE integrator
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
Listed in 2020 Dean's List of Exceptional ThesesCopyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author.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 -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
- …