Learning seminar on contact Hamiltonian systems

Seminar Description

The purpose of this seminar is to study basic notions of contact geometry, and contact Hamiltonian systems.  Hamilton's equations are more known in the context of symplectic geometry, where they model conservative mechanical systems. Through contact geometry, we will see how to account for systems interacting with the environment. 

We will study the relation between contact and symplectic manifolds, and how contact manifolds provide a good framework for an Hamiltonian formulation of time-dependant and dissipative systems. Further topics may include Jacobi manifolds, their relation to Poisson manifolds, Hamiltonian systems on them, and modern approaches to contact Hamilton-Jacobi theory.

Seminar organization

The format is that of a learning seminar - the material will be presented by participants of the seminar; although the speaker of each session prepares the material in advance, the idea is to keep the environment welcoming (particularly to students), having in mind that the speaker and the audience are learning together.

The first sessions are scheduled for April 4 and 11, 2024. The following sessions, and their topics, will be scheduled together by the participants. The schedule of the seminar will be posted and updated on this webpage.

Prerequisites

Basic knowledge from differential geometry (manifolds, differential forms).

Schedule

Where: Room 5.5

1. Introduction to contact manifolds and contact Hamiltonian systems (April 4, 14:30-16:30)  - Gonçalo Oliveira 

2. Introduction to contact manifolds and contact Hamiltonian systems, part II (April 11, 14:30-16:30)  - Gonçalo Oliveira

3. Time-dependent contact Hamiltonian systems (April 16, 14:30-16:30)  - Gonçalo Oliveira

4. Hamilton-Jacobi formulation of contact Hamiltonian systems (April 23, 14:30-16:30)  - Gonçalo Oliveira

5. Hamilton-Jacobi theory and examples (May 2, 14:30-16:30)  - Gonçalo Oliveira / João Nuno Mestre

6. Hamilton-Jacobi theory through examples - oscillators (May 9, 15:30-17:30)  - João Nuno Mestre

7. Hamilton-Jacobi theory through examples - damped oscillators (May 16, 15:30-17:30)  - Lennart Obster

8. Hamilton-Jacobi theory through examples - damped parametric oscillators (May 29, 16:00-17:00)  - Gonçalo Oliveira

9. To be decided... 

Bibliography

A. Bravetti, H. Cruz, D. Tapias, Contact Hamiltonian Mechanics, 2016.

Complementary references

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer-Verlag, 2001 and 2008 (corrected printing). (available on the author's webpage).
For convenience, here is the errata, in two versions: for the Springer 2008 printed text and for the author's 2006 website text (updated July 2021).

V. I. Arnold, Mathematical Methods of Classical Mechanics, second edition, 1989.  https://link.springer.com/book/10.1007/978-1-4757-2063-1

P. Libermann, C.-M. Marle, Symplectic geometry and analytical mechanics, 1987. https://link.springer.com/chapter/10.1007/978-94-009-3807-6_5

K. Grabowska, J. Grabowski, A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory.