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

Abstract

We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact manifold M itself but sections of a line bundle over M or, equivalently, 1-homogeneous functions on a certain GL(1,R)-principal bundle P\to M which is equipped with a homogeneous symplectic form ω. In other words, our understanding of contact geometry is that it is not an `odd-dimensional cousin' of symplectic geometry but rather a part of the latter, namely `homogeneous symplectic geometry'. This understanding of contact structures is much simpler than the traditional one and very effective in applications, reducing the contact Hamiltonian formalism to the standard symplectic picture. We develop in this language contact Hamiltonian mechanics in autonomous, as well as time-dependent case, and the corresponding Hamilton-Jacobi theory. Fundamental examples are based on canonical contact structures on the first jet bundles J^1(L) of sections of line bundles L, which play in contact geometry a fundamental role similar to that played by cotangent bundles in symplectic geometry.

arXiv comment31 pages

References in corpus (35)

• Contact Hamiltonian Mechanics
• Higher vector bundles and multi-graded symplectic manifolds
• Convergence of the solutions of the discounted equation
• Convergence of the solutions of the discounted equation: the discrete case
• Tangent Lifts of Poisson and Related Structures
• Contact Hamiltonian Systems
• Dirac Algebroids in Lagrangian and Hamiltonian Mechanics
• Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems
• Geometry of Thermodynamic Processes
• New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries
• Jacobi structures revisited
• A Hamilton-Jacobi Formalism for Thermodynamics
• Graded contact manifolds and contact Courant algebroids
• Graded bundles and homogeneity structures
• Contact manifolds and dissipation, classical and quantum
• The graded Jacobi algebras and (co)homology
• Variational calculus with constraints on general algebroids
• Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics
• Quasi-derivations and QD-algebroids
• Remarks on Contact and Jacobi Geometry
• Linear duals of graded bundles and higher analogues of (Lie) algebroids
• A Hamiltonian approach to Thermodynamics
• Brackets
• Modular classes of skew algebroid relations
• Higher order mechanics on graded bundles
• On the role of geometry in statistical mechanics and thermodynamics I: Geometric perspective
• On the role of geometry in statistical mechanics and thermodynamics II: Thermodynamic perspective
• Unified Lagrangian-Hamiltonian formalism for contact systems
• Differential graded contact geometry and Jacobi structures
• Extended Hamilton-Jacobi theory, contact manifolds and integrability by quadratures
• Tulczyjew Triples in Higher Derivative Field Theory
• Tulczyjew triples: from statics to field theory
• VB-structures and generalizations
• Smooth subsolutions of the discounted Hamilton-Jacobi equations
• Implicit Contact Dynamics and Hamilton-Jacobi Theory

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• Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems
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• Contact Lie systems
• Nonautonomous k-contact field theories
• The Herglotz variational principle for dissipative field theories
• Lie integrability by quadratures for symplectic, cosymplectic, contact and cocontact Hamiltonian systems
• A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian
• Contactifications: a Lagrangian description of compact Hamiltonian systems
• Practical Introduction to Action-Dependent Field Theories
• Cosymplectic geometry, reductions, and energy-momentum methods with applications
• Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
• Brackets in multicontact geometry and multisymplectization
• Equilibria and bifurcations in contact dynamics
• Jacobi algebroids and Jacobi sigma models
• Relative Equilibria for Scaling Symmetries and Central Configurations