Contact Geometry in Hamiltonian Dynamics
Summary
Contact geometry extends the symplectic framework to odd-dimensional phase spaces, providing a natural arena for systems with dissipation or energy exchange. A contact manifold is endowed with a one-form whose maximal non-integrability encodes irreversible processes, while its associated Reeb and Hamiltonian vector fields govern evolution. Contact Hamiltonian dynamics generalises conservative motion by allowing the Hamiltonian function to drive both energy-conserving and dissipative components. This structure underpins a unified geometric treatment of thermodynamics, statistical mechanics and mechanical systems with friction or thermal interactions. Key innovations include the Herglotz variational principle, in which the action evolves according to contact rules, and symplectification techniques that embed contact manifolds into higher-dimensional symplectic cones. Such methods facilitate the characterisation of equilibrium states, entropy production and non-equilibrium trajectories, and support modern developments in optimal control, where dissipation plays a central role.
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Contact Geometry in Hamiltonian Dynamics publication trend
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Technical terms
Contact manifold: An odd-dimensional differentiable manifold equipped with a one-form whose maximal non-integrability defines the contact structure.
Contact form: A one-form θ satisfying θ∧(dθ)^n≠0 everywhere, which determines the hyperplane distribution of a contact manifold.
Reeb vector field: The unique vector field R on a contact manifold such that θ(R)=1 and dθ(R,·)=0, characterising the flow transverse to the contact distribution.
Contact Hamiltonian: A function H on a contact manifold whose associated vector field X_H obeys i_{X_H}dθ=dH−(R·H)θ and θ(X_H)=−H, combining conservative and dissipative effects.
Herglotz variational principle: A generalisation of the classical variational principle in which the action evolves along trajectories via a contact Hamiltonian equation, accommodating dissipation.
Cocontact manifold: A manifold with two compatible one-forms structured to describe explicitly time-dependent contact systems in both Hamiltonian and Lagrangian formalisms.
Dissipated quantity: A function on a contact manifold that decreases or increases monotonically along contact Hamiltonian flows, generalising conserved quantities in symplectic dynamics.
References
- Symmetries, Conservation and Dissipation in Time‐Dependent Contact Systems. Fortschritte der Physik (2023).
- Contact Hamiltonian Dynamics: The Concept and Its Use. Entropy (2017).
- Optimal Control, Contact Dynamics and Herglotz Variational Problem. Journal of Nonlinear Science (2022).
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