New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries

Abstract

We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian formalisms, studying their equivalence. We define several kinds of symmetries for contact dynamical systems, as well as the notion of dissipation laws, prove a dissipation theorem and give a way to construct conserved quantities. Some well-known examples of dissipative systems are discussed.

24 pp. This is the final version of the article. Some minor misprints are corrected. Some bibliographycal references are uploaded

References in corpus (14)

• The classical mechanics of non-conservative systems
• Contact Hamiltonian Mechanics
• A survey on cosymplectic geometry
• Contact Hamiltonian Systems
• Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems
• Contact manifolds and dissipation, classical and quantum
• Infinitesimal symmetries in Contact Hamiltonian systems
• A contact geometry framework for field theories with dissipation
• Singular Lagrangians and precontact Hamiltonian Systems
• Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics
• On the k-Symplectic, k-Cosymplectic and Multisymplectic Formalisms of Classical Field Theories
• On the lagrangian description of dissipative systems
• Symmetries and infinitesimal symmetries of singular differential equations
• Fibre derivatives: some applications to singular lagrangians

Cited by in corpus (31)

• Invariant measures for contact Hamiltonian systems: symplectic sandwiches with contact bread
• A $k$-contact Lagrangian formulation for nonconservative field theories
• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Multicontact formulation for non-conservative field theories
• On the geometry of discrete contact mechanics
• Numerical integration in celestial mechanics: a case for contact geometry
• Unified Lagrangian-Hamiltonian formalism for contact systems
• Reviewing the Geometric Hamilton-Jacobi Theory concerning Jacobi and Leibniz identities
• 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
• Reductions: precontact versus presymplectic
• Nonautonomous k-contact field theories
• Contact Lie systems
• Contact Lagrangian systems subject to impulsive constraints
• The Herglotz variational principle for dissipative field theories
• Application of Herglotz's Variational Principle to Electromagnetic Systems with Dissipation
• Lie integrability by quadratures for symplectic, cosymplectic, contact and cocontact Hamiltonian systems
• Inverse problem and equivalent contact systems
• Nonsmooth Herglotz variational principle
• A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian
• On Locally Conformally Cosymplectic Hamiltonian Dynamics and Hamilton-Jacobi Theory
• Generalized virial theorem for contact Hamiltonian systems
• Practical Introduction to Action-Dependent Field Theories
• The flow method for the Baker-Campbell-Hausdorff formula: exact results
• Invariant measures for some dissipative systems from the Jacobi last multiplier
• Brackets in multicontact geometry and multisymplectization
• When action is not least for systems with action-dependent Lagrangians
• Constrained Lagrangian dissipative contact dynamics
• Skinner-Rusk formalism for k-contact systems
• Geometric numerical integration of Liénard systems via a contact Hamiltonian approach