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.

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

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• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
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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