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Contact geometry in infinite dimensions

Fraser Aidan Kelvin Sanders

TL;DR

The paper develops a coherent framework for infinite-dimensional contact geometry by defining weakly contact structures on manifolds modelled by convenient spaces, thereby extending dissipation-compatible Hamiltonian mechanics beyond finite dimensions. It provides a thorough background, introduces multiple equivalent characterisations (including a generalized De Rham-style flat map $\flat$) and proves a key property $\ η \wedge (dη)^k \neq 0$ for all $k$, which supports a Pfaffian-type theory in the infinite-dimensional setting. It then constructs concrete infinite-dimensional contact manifolds (via products and jet-bundle methods), derives Hamiltonian dynamics on Fréchet spaces, and presents explicit examples such as a damped oscillator and a damped wave equation with a source term, using both cosymplectic and cocontract structures to model time dependence and dissipation. The work culminates with cosymplectic and cocontact generalisations, including a Darboux-type result and explicit wave-equation-with-source formulations, thereby providing mathematical tools to study dissipative PDEs and time-dependent Hamiltonian systems in infinite dimensions with potential physical applications.

Abstract

We generalise the theories of cosymplectic, contact, and cocontact manifolds to the infinite-dimensional setting and calculate model examples of time-dependent and dissipative Hamiltonian systems.

Contact geometry in infinite dimensions

TL;DR

The paper develops a coherent framework for infinite-dimensional contact geometry by defining weakly contact structures on manifolds modelled by convenient spaces, thereby extending dissipation-compatible Hamiltonian mechanics beyond finite dimensions. It provides a thorough background, introduces multiple equivalent characterisations (including a generalized De Rham-style flat map ) and proves a key property for all , which supports a Pfaffian-type theory in the infinite-dimensional setting. It then constructs concrete infinite-dimensional contact manifolds (via products and jet-bundle methods), derives Hamiltonian dynamics on Fréchet spaces, and presents explicit examples such as a damped oscillator and a damped wave equation with a source term, using both cosymplectic and cocontract structures to model time dependence and dissipation. The work culminates with cosymplectic and cocontact generalisations, including a Darboux-type result and explicit wave-equation-with-source formulations, thereby providing mathematical tools to study dissipative PDEs and time-dependent Hamiltonian systems in infinite dimensions with potential physical applications.

Abstract

We generalise the theories of cosymplectic, contact, and cocontact manifolds to the infinite-dimensional setting and calculate model examples of time-dependent and dissipative Hamiltonian systems.

Paper Structure

This paper contains 25 sections, 16 theorems, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Topological vector spaces
  4. Smooth curves and convenient spaces
  5. Fréchet spaces
  6. Banach spaces
  7. Differential geometry in infinite dimensions
  8. Symplectic manifolds
  9. Infinite-dimensional contact manifolds
  10. Examples and constructions
  11. Contact manifolds via products
  12. Canonical contact structures beyond Banach spaces
  13. Submanifolds of contact type
  14. Contact Hamiltonian dynamics
  15. Hamiltonian vector fields
  16. ...and 10 more sections

Key Result

Theorem 3.2

Let $M$ be an infinite-dimensional manifold, and $\eta\in \Omega^1(M)$ be a 1-form such that $d\eta$ is degenerate (has non-zero kernel). We define the horizontal and vertical distributions (subbundles of $TM$) as $\mathcal{H}:= \ker \eta$ and $\mathcal{V}:= \ker d \eta = \ker (v \mapsto d\eta(v,-))

Figures (1)

  • Figure :

Theorems & Definitions (75)

  • Definition 2.1: TVS and lcTVS
  • Definition 2.2: smooth maps
  • Definition 2.3: Convenient vector space
  • Definition 2.4: seminorm
  • Definition 2.5: Fréchet space
  • Definition 2.6: Banach space
  • Definition 2.7: Manifold
  • Definition 2.8
  • Definition 2.9: Kinematic tangent bundle
  • Definition 2.10: Kinematic cotangent bundle
  • ...and 65 more