First Steps Towards a Symplectic Dynamics

TL;DR

This paper explores the integration of dynamical systems and symplectic geometry to develop a unified approach to Hamiltonian systems, highlighting foundational problems at their intersection.

Contribution

It initiates the development of a combined framework merging dynamical systems and symplectic geometry for Hamiltonian systems.

Findings

Identification of key problems at the intersection of the two fields

Proposals for integrating geometric and dynamical ideas

Foundational steps towards a unified theory

Abstract

Many interesting physical systems have mathematical descriptions as finite-dimensional or infinite-dimensional Hamiltonian systems. Poincare who started the modern theory of dynamical systems and symplectic geometry developed a particular viewpoint combining geometric and dynamical systems ideas in the study of Hamiltonian systems. After Poincare the field of dynamical systems and the field of symplectic geometry developed separately. Both fields have rich theories and the time seems ripe to develop the common core with highly integrated ideas from both fields. We discuss problems which show how dynamical systems questions and symplectic ideas come together in a nontrivial way.