A contact geometry framework for field theories with dissipation

Abstract

We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of $k$-contact structure and $k$-contact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the $k$-symplectic Hamiltonian systems in field theory. The concepts of symmetries and dissipation laws are introduced and developed. Two relevant examples are analyzed in detail: the damped vibrating string and Burgers' equation.

arXiv comment26 pp. Minor corrections. The bibliography has been updated

References in corpus (6)

• Contact Hamiltonian Mechanics
• Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems
• Contact manifolds and dissipation, classical and quantum
• On the k-Symplectic, k-Cosymplectic and Multisymplectic Formalisms of Classical Field Theories
• Symmetries and conservation laws in the Gunther k-symplectic formalism of field theory
• Symmetries and infinitesimal symmetries of singular differential equations

Cited by in corpus (24)

• New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries
• Invariant measures for contact Hamiltonian systems: symplectic sandwiches with contact bread
• A $k$-contact Lagrangian formulation for nonconservative field theories
• Multicontact formulation for non-conservative field theories
• Numerical integration in celestial mechanics: a case for contact geometry
• Unified Lagrangian-Hamiltonian formalism for contact systems
• Symmetries, conservation and dissipation in time-dependent contact systems
• Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems
• Higher-order contact mechanics
• Contact Lie systems
• Nonautonomous k-contact field theories
• Contact Lagrangian systems subject to impulsive constraints
• Application of Herglotz's Variational Principle to Electromagnetic Systems with Dissipation
• The Herglotz variational principle for dissipative field theories
• An overview of the Hamilton--Jacobi theory: the classical and geometrical approaches and some extensions and applications
• A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian
• Inverse problem and equivalent contact systems
• Practical Introduction to Action-Dependent Field Theories
• Brackets in multicontact geometry and multisymplectization
• Constrained Lagrangian dissipative contact dynamics
• Skinner-Rusk formalism for k-contact systems
• When action is not least for systems with action-dependent Lagrangians
• Reduction and reconstruction of multisymplectic Lie systems
• Geometric numerical integration of Liénard systems via a contact Hamiltonian approach