New Directions for Contact Integrators

Abstract

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.

Orbital dynamics on invariant sets of contact Hamiltonian systems

“…From the point of view of mathematics, there are two routes to study dynamics of contact Hamiltonian systems, including the framework of contact geometry and the method of differential equation and dynamical system. On one hand, for example, the geometric formulation of the Lagrangian and Hamiltonian formalisms 5822 QIHUAI LIU AND PEDRO J. TORRES of dissipative mechanical systems has been studied by using the tool of contact geometry in a recent work [16,7], where a new expression of the dynamical equations of contact Hamiltonian system is established without using the Reeb vector field. On the other hand, the research of dynamics of contact Hamiltonian system usually includes the study of its variational principles, Aubry-Mather theory, periodic and quasi-periodic orbits and invariant measure, see [9,31,34] for instance.…”