Time-dependent contact mechanics

Abstract

Contact geometry allows us to describe some thermodynamic and dissipative systems. In this paper we introduce a new geometric structure in order to describe time-dependent contact systems: cocontact manifolds. Within this setting we develop the Hamiltonian and Lagrangian formalisms, both in the regular and singular cases. In the singular case, we present a constraint algorithm aiming to find a submanifold where solutions exist. As a particular case we study contact systems with holonomic time-dependent constraints. Some regular and singular examples are analyzed, along with numerical simulations.

Nonautonomous k-contact field theories

“…The conditions in Definition 3.1 mean the following: (1) η = 0 everywhere, (2) τ = 0 everywhere, (4) τ ∧ η = 0, (5) ker τ ∩ ker η ∩ ker dη = {0}, which implies that ker dη has rank 0, 1 or 2, and (3) implies that ker dη has rank 2. Thus, a 1-cocontact structure coincides with the cocontact structure introduced in [12] to describe time-dependent contact mechanics.…”

“…Remark 5.13. In the case k = 1, we recover the cocontact Lagrangian formalism presented in the recent paper [12] for time-dependent contact Lagrangian systems.…”

“…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]. It is worth pointing out that contact geometry allows to study more systems than just dissipative ones [23].…”

Lie integrability for time-independent and time-dependent Hamiltonian systems

“…Finally we completes the notation and language employed in this paper. Here we briefly review the formalism of time-dependent contact Hamiltonian systems introduced in [26].…”

Symmetries, Conservation and Dissipation in Time‐Dependent Contact Systems

“…When a classical mechanical system exhibits explicit time dependence, i.e., it is non-autonomous, its underlying DOI: 10.1002/prop.202300048 geometric structure can be taken either as a contact structure or as a cosymplectic structure. [22] Recently, the so-called cocontact geometry, [23,24] a suitable geometric structure describing non-autonomous dissipative systems, combining contact and cosymplectic geometry, has been introduced.…”