Skinner–Rusk formalism for k-contact systems

Gràcia, Xavier; Rivas, Xavier; Román‐Roy, Narciso

doi:10.1016/j.geomphys.2021.104429

Cited by 11 publications

(15 citation statements)

References 41 publications

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“…It is worth pointing out that contact geometry allows to study more systems than just dissipative ones [23]. In the last years, a generalization of both contact and k-symplectic structures was devised to describe autonomous field theories with damping [24,26,32] both in the Hamiltonian and Lagrangian formulations.…”

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confidence: 99%

Nonautonomous k-contact field theories

Rivas¹

2022

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This paper provides a new geometric framework to describe non-conservative field theories with explicit dependence on the space-time coordinates by combining the k-cosymplectic and k-contact formulations. This geometric framework, the k-cocontact geometry, permits to develop a Hamiltonian and Lagrangian formalisms for these field theories. We also compare this new formulation in the autonomous case with the previous k-contact formalism. To illustrate the theory, we study the nonlinear damped wave equation with external timedependent forcing.

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confidence: 99%

Nonautonomous k-contact field theories

Rivas¹

2022

Preprint

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“…Recently, the contact formulation for non-conservative mechanical systems has been generalised via the so-called k-contact [40,42,55], k-cocontact [69], and multicontact [27,80] formulations. It would be interesting to study the Lie systems whose VG Lie algebra consists of Hamiltonian vector fields relative to these structures.…”

Section: Conclusion and Further Researchmentioning

confidence: 99%

Contact Lie systems: theory and applications

Lucas

Rivas

2023

J. Phys. A: Math. Theor.

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A Lie system is a time-dependent system of differential equations describing the integral curves of a time-dependent vector field that can be considered as a curve in a finite-dimensional Lie algebra of vector fields $V$. We call $V$ a Vessiot--Guldberg Lie algebra.We define and analyse contact Lie systems, namely Lie systems admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a contact manifold. We also study contact Lie systems of Liouville type, which are invariant relative to the flow of a Reeb vector field. Liouville theorems, contact Marsden--Weinstein reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Contact Lie systems on three-dimensional Lie groups with Vessiot--Guldberg Lie algebras of right-invariant vector fields and associated with left-invariant contact forms are classified. Our results are illustrated by examples with relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, quantum mechanical systems, etc. Finally, a Poisson coalgebra method to derive superposition rules for contact Lie systems of Liouville is developed.

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“…The formulation presented in this work is also a first step towards finding a geometric formalism for non-autonomous dissipative field theories based on the k-contact setting [25,27,34] and generalizing the multisymplectic formalism [12,46]. The k-contact formalism allows to describe autonomous field theories, such as field theories with damping, some equations from circuit theory, such as the socalled telegrapher's equation, or the Burgers' equation.…”

Section: Conclusion and Further Researchmentioning

confidence: 99%

Time-dependent contact mechanics

León¹,

Gaset²,

Gràcia³

et al. 2022

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Contact geometry allows 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.

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Skinner–Rusk formalism for k-contact systems

Cited by 11 publications

References 41 publications

Nonautonomous k-contact field theories

Nonautonomous k-contact field theories

Contact Lie systems: theory and applications

Time-dependent contact mechanics

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