arXiv subjects

Xavier Rivas

Publications and source records attributed to Xavier Rivas.

31 records · Page 2Linked to original sources

arXiv2023

Time-dependent contact mechanics

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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arXiv2023

Multicontact formulation for non-conservative field theories

A new geometric framework is developed to describe non-conservative classical field theories, which is based on multisymplectic and contact geometries. Assuming certain additional conditions and using the forms that define this multicontact structure, as well as other geometric elements that are derived from them, we can introduce variational field equations in the multicontact manifolds. These equations are stated using different geometric tools; namely, sections, multivector fields and Ehresmann connections in fiber bundles. Then, this framework can be adapted to the jet bundle description of classical field theories and the field equations are stated both in the Lagrangian and the Hamiltonian formalisms, which are discussed in the regular and the singular cases.

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arXiv2022

The Herglotz variational principle for dissipative field theories

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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arXiv2022

Contact Lie systems

We define and analyse the properties of contact Lie systems, namely systems of first-order differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of Hamiltonian vector fields relative to a contact structure. As a particular example, we study families of conservative contact Lie systems. Liouville theorems, contact reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Our results are illustrated by examples with relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, etcetera.

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arXiv2022

Nonautonomous k-contact field theories

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 time-dependent forcing.

math-ph↗

arXiv2022

Reduction and reconstruction of multisymplectic Lie systems

A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. In this work, multisymplectic structures are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie systems. Conversely, we study how reduced multisymplectic Lie systems allow us to retrieve the form of the multisymplectic Lie system that gave rise to them. Our results are illustrated with examples occurring in physics, mathematics, and control theory.

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arXiv2022

Lagrangian-Hamiltonian formalism for time-dependent dissipative mechanical systems

In this paper we present a unified Lagrangian--Hamiltonian geometric formalism to describe time-dependent contact mechanical systems, based on the one first introduced by K. Kamimura and later formalized by R. Skinner and R. Rusk. This formalism is especially interesting when dealing with systems described by singular Lagrangians, since the second-order condition is recovered from the constraint algorithm. In order to illustrate this formulation, some relevant examples are described in full detail: the Duffing equation, an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.

math-ph↗

arXiv2022

Skinner-Rusk formalism for k-contact systems

In previous papers, a geometric framework has been developed to describe non-conservative field theories as a kind of modified Lagrangian and Hamiltonian field theories. This approach is that of $k$-contact Hamiltonian systems, which is based on the $k$-symplectic formulation of field theories as well as on contact geometry. In this work we present the Skinner--Rusk unified setting for these kinds of theories, which encompasses both the Lagrangian and Hamiltonian formalisms into a single picture. This unified framework is specially useful when dealing with singular systems, since: (i) it incorporates in a natural way the second-order condition for the solutions of field equations, (ii) it allows to implement the Lagrangian and Hamiltonian constraint algorithms in a unique simple way, and (iii) it gives the Legendre transformation, so that the Lagrangian and the Hamiltonian formalisms are obtained straightforwardly. We apply this description to several interesting physical examples: the damped vibrating string, the telegrapher's equations, and Maxwell's equations with dissipation terms.

math-ph↗

arXiv2021

Constraint algorithm for singular field theories in the $k$-cosymplectic framework

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of $k$-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of $k$-precosymplectic structure, which is a generalization of the $k$-cosymplectic structure. Next $k$-precosymplectic Hamiltonian systems are introduced in order to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a submanifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.

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arXiv2020

Unified Lagrangian-Hamiltonian formalism for contact systems

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of contact autonomous mechanical systems, which is based on the approach of the pionnering work of R. Skinner and R. Rusk. This framework permits to skip the second order differential equation problem, which is obtained as a part of the constraint algorithm (for singular or regular Lagrangians), and is specially useful to describe singular Lagrangian systems. Some examples are also discussed to illustrate the method.

math-ph↗

arXiv2020

New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries

We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian formalisms, studying their equivalence. We define several kinds of symmetries for contact dynamical systems, as well as the notion of dissipation laws, prove a dissipation theorem and give a way to construct conserved quantities. Some well-known examples of dissipative systems are discussed.

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arXiv2020

A $k$-contact Lagrangian formulation for nonconservative field theories

We present a geometric Lagrangian formulation for first-order field theories with dissipation. This formulation is based on the $k$-contact geometry introduced in a previous paper, and gathers contact Lagrangian mechanics with $k$-symplectic Lagrangian field theory together. We also study the symmetries and dissipation laws for these nonconservative theories, and analyze some examples.

math-ph↗

arXiv2020

A contact geometry framework for field theories with dissipation

We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of $k$-contact structure and $k$-contact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the $k$-symplectic Hamiltonian systems in field theory. The concepts of symmetries and dissipation laws are introduced and developed. Two relevant examples are analyzed in detail: the damped vibrating string and Burgers' equation.

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