Infinite-dimensional contact geometry and conservative contact systems
Speaker: Fraser Sanders (University of Manchester)
Abstract: In finite dimensions, contact geometry is well-studied as an odd-dimensional counterpart to symplectic geometry. It is also a natural setting in which to generalise Hamiltonian mechanics of systems, taking into account a notion of dissipation or diffusion. For example, the simplest example of a contact-Hamiltonian system is the damped harmonic oscillator (the dissipative counterpart to the simple harmonic oscillator). Due to the diffusive properties of the contact-Hamilton equations, it is also a natural setting in which to study thermodynamics (cf. Grmela, Bravetti, and many others). In fact, the 1st law of thermodynamics, $dU = T dS$ can be considered as defining a contact form on $\mathbb{R}^3$.
The goal during the seminar is to show that contact geometry can be meaningfully defined in infinite-dimensional manifolds, and to give some elementary examples of infinite-dimensional contact Hamiltonian systems.
Time permitting, I will also discuss some subset of:Â
- Interesting differences between finite- and infinite-dimensional contact geometry,
- Infinite-dimensional counterparts of Hopf fibrations,Â
- And how to define a large class of conservative contact systems, and some examples of this concept.