A geometric 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 τ : P → 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 symplec-Jtic geometry’. This understanding of contact structures is much simpler than the traditionalone and very effective in applications, reducing the contact Hamiltonian formalism to thestandard symplectic picture. We develop in this language contact Hamiltonian mechanics inautonomous, as well as time-dependent case, and the corresponding Hamilton-Jacobi theory.Fundamental examples are based on canonical contact structures on the first jet bundles1(L) of sections of line bundles L, which play in contact geometry a fundamental rˆole, similar to that played by cotangent bundles in symplectic geometry.

Nonautonomous k-contact field theories

“…In particular, contact geometry [4,27,33] has been used to study mechanical systems with dissipation [5,7,10,14,25,38]. This has many applications in thermodynamics [6,43], quantum mechanics [11], circuit theory [28] and control theory [39] among others [8,16,15,17,29,30]. Recently, contact mechanics have been generalized in order to deal with time-dependent contact systems [12,41].…”

Scaling symmetries, contact reduction and Poincaré’s dream

“…Among these, the most striking result is having re-obtained McGehee's blow-up for collisions in n-body systems in terms of a contact reduction and, therefore, as a Λ-Herglotz system (proposition 9). For future work, a neat and general framework for the realization of our contact reduction is the one put forward in [28,29]. Thus we expect to analyze some of the results presented here within this context.…”

“…For comparison to standard constructions in projective geometry and the 's-scalars or vector fields' used in [1], as well as the structure used in [28] and [29], one may state all our results here in terms of appropriate associated bundles over C, whose sections correspond to degree α functions, vector fields, one-forms, etc on M under D. Our scaling functions (and their resulting coordinate descriptions) correspond then to working in a local trivialization of these bundles. In what follows however, we will simply use scaling functions on M to obtain explicit expressions.…”

“…The authors would like to thank Mats Vermeeren for stimulating questions and Janusz Grabowski for pointing out [28,29], as well as Alain Albouy for pointing out [12,13]. The work of A Bravetti was partially supported by DGAPA-UNAM, program PAPIIT, Grant No.…”

Singularities, Bifurcations and Catastrophes

“…A further approach is taken by Grabowska and Grabowski [13]; they explicitly put ξ at the forefront by considering the line bundle ξ • ⊂ T * M whose fibre is the annihilator of ξ.…”