Contact Lie systems

Abstract

We define and analyse the properties of contact Lie systems, namely systems of first-order differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of Hamiltonian vector fields relative to a contact structure. As a particular example, we study families of conservative contact Lie systems. Liouville theorems, contact reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Our results are illustrated by examples with relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, etcetera.

arXiv comment29 pp, 4 figures. New version of the manuscript with Sections 4, 5.4, and 6 added. Many new results included and typos corrected

References in corpus (41)

• Contact Hamiltonian Mechanics
• A survey on cosymplectic geometry
• Contact Hamiltonian Systems
• Superposition rules, Lie theorem and partial differential equations
• Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems
• New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries
• Lie systems: theory, generalisations, and applications
• Contact manifolds and dissipation, classical and quantum
• Infinitesimal symmetries in Contact Hamiltonian systems
• A contact geometry framework for field theories with dissipation
• Lie-Hamilton systems on the plane: properties, classification and applications
• Singular Lagrangians and precontact Hamiltonian Systems
• A $k$-contact Lagrangian formulation for nonconservative field theories
• Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics
• From constants of motion to superposition rules for Lie-Hamilton systems
• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Lie--Hamilton systems: theory and applications
• Dirac--Lie systems and Schwarzian equations
• Brackets
• Superposition rules for higher-order systems and their applications
• Lie-Hamilton systems on the plane: Applications and superposition rules
• k-symplectic Lie systems: theory and applications
• Multicontact formulation for non-conservative field theories
• Periodic first integrals for Hamiltonian systems of Lie type
• Reduction of multisymplectic manifolds
• Unified Lagrangian-Hamiltonian formalism for contact systems
• Symmetries, conservation and dissipation in time-dependent contact systems
• L-infinity Algebras From Multicontact Geometry
• Mixed superposition rules and the Riccati hierarchy
• On Lie systems and Kummer-Schwarz equations
• Nonautonomous k-contact field theories
• Reductions: precontact versus presymplectic
• Generalized nonconservative gravitational field equations from Herglotz action principle
• Remarks on multisymplectic reduction
• Multisymplectic structures and invariant tensors for Lie systems
• Jacobi-Lie Hamiltonian systems on real low-dimensional Jacobi-Lie groups and their Lie symmetries
• A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian
• Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi-Lie systems
• Geometric features of Vessiot--Guldberg Lie algebras of conformal and Killing vector fields on ${\mathbb{R}}^2$
• Geometrical aspects of contact mechanical systems and field theories
• Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras

Cited by in corpus (8)

• Multicontact formulation for non-conservative field theories
• Nonautonomous k-contact field theories
• The Herglotz variational principle for dissipative field theories
• A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$
• Cosymplectic geometry, reductions, and energy-momentum methods with applications
• Quasi-rectifiable Lie algebras for partial differential equations
• Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
• New Lie systems from Goursat distributions: reductions and reconstructions