Abstract
AbstractWe 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 manifoldMitself but sections of a line bundle overMor, equivalently, 1-homogeneous functions on a certainGL(1,R)-principal bundleτ:P→M, which is equipped with a homogeneous symplectic formω. 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 the autonomous, as well as the time-dependent case, and the corresponding Hamilton–Jacobi theory. Fundamental examples are based on canonical contact structures on the first jet bundlesJ1Lof sections of line bundlesL, which play in contact geometry a fundamental rôle, similar to that played by cotangent bundles in symplectic geometry.
Subject
General Physics and Astronomy,Mathematical Physics,Modeling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics
Cited by
8 articles.
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1. Scaling symmetries, contact reduction and Poincaré’s dream;Journal of Physics A: Mathematical and Theoretical;2023-10-09
2. Implicit contact dynamics and Hamilton-Jacobi theory;Differential Geometry and its Applications;2023-10
3. Contact Lie systems: theory and applications;Journal of Physics A: Mathematical and Theoretical;2023-07-28
4. Thermodynamic Entropy as a Noether Invariant from Contact Geometry;Entropy;2023-07-19
5. Reductions: precontact versus presymplectic;Annali di Matematica Pura ed Applicata (1923 -);2023-06-02