Nonautonomous <i>k</i>-contact field theories

Rivas, Xavier

doi:10.1063/5.0131110

Cited by 3 publications

(1 citation statement)

References 38 publications

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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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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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Symmetries, Conservation and Dissipation in Time‐Dependent Contact Systems

Gaset

Asier

Rivas

2023

Fortschritte der Physik

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In contact Hamiltonian systems, the so‐called dissipated quantities are akin to conserved quantities in classical Hamiltonian systems. In this article, a Noether's theorem for non‐autonomous contact Hamiltonian systems is proved, characterizing a class of symmetries which are in bijection with dissipated quantities. Other classes of symmetries which preserve (up to a conformal factor) additional structures, such as the contact form or the Hamiltonian function, are also studied. Furthermore, making use of the geometric structures of the extended tangent bundle, additional classes of symmetries for time‐dependent contact Lagrangian systems are introduced. The results are illustrated with several examples. In particular, the two‐body problem with time‐dependent friction is presented, which could be interesting in celestial mechanics.

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The Herglotz variational principle for dissipative field theories

Gaset,

Lainz,

Mas

et al. 2024

Geom. Mech.

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In the recent years, with the incorporation of contact geometry, there has been a renewed interest in the study of dissipative or non-conservative systems in physics and other areas of applied mathematics. The equations arising when studying contact Hamiltonian systems can also be obtained via the Herglotz variational principle. The contact Lagrangian and Hamiltonian formalisms for mechanical systems has also been generalized to field theories. The main goal of this paper is to develop a generalization of the Herglotz variational principle for first-order and higher-order field theories. In order to illustrate this, we study three examples: the damped vibrating string, the Korteweg–de Vries equation, and an academic example showing that the non-holonomic and the vakonomic variational principles are not fully equivalent.

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Nonautonomous k-contact field theories

Cited by 3 publications

References 38 publications

Contact Lie systems: theory and applications

Contact Lie systems: theory and applications

Symmetries, Conservation and Dissipation in Time‐Dependent Contact Systems

The Herglotz variational principle for dissipative field theories

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