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Modesto Salgado

Publications and source records attributed to Modesto Salgado.

14 recordsLinked to original sources
arXiv2024

Symmetry reduction and reconstruction in contact geometry and Lagrange-Poincar\'e-Herglotz equations

In this paper, we investigate the reduction process of a contact Lagrangian system whose Lagrangian is invariant under a group of symmetries. We give explicit coordinate expressions of the resulting reduced differential equations, the so-called Lagrange-Poincare-Herglotz equations. Our framework relied on the associated Herglotz vector field and its projected vector field, and the use of well-chosen quasi-velocities. Some examples are also discussed.

math-ph↗
arXiv2023

Contact formalism for dissipative mechanical systems on Lie algebroids

In this paper, we introduce a geometric description of contact Lagrangian and Hamiltonian systems on Lie algebroids in the framework of contact geometry, using the theory of prolongations. We discuss the relation between Lagrangian and Hamiltonian settings through a convenient notion of Legendre transformation. We also discuss the Hamilton-Jacobi problem in this framework and introduce the notion of a Legendrian Lie subalgebroid of a contact Lie algebroid.

math.SG↗
arXiv2023

Jacobi structure for dissipative mechanical systems on Lie Algebroids

We extend the Jacobi structure from $TQ\times \mathbb{R}$ and $T^{*}Q \times \mathbb{R}$ to $A\times \mathbb{R}$ and $A^{*}\times \mathbb{R}$, respectively, where $A$ is a Lie algebroid and $A^{*}$ carries the associated Poisson structure. We see that $A^*\times \mathbb{R}$ possesses a natural Jacobi structure from where we are able to model dissipative mechanical systems, generalizing previous models on $TQ\times \mathbb{R}$ and $\mathfrak{g}\times \mathbb{R}$.

math-ph↗
arXiv2023

Some contributions to k-contact Lagrangian field equations, symmetries and dissipation laws

It is well known that k-contact geometry is a suitable framework to deal with non-conservative field theories. In this paper, we study some relations between solutions of the k-contact Euler-Lagrange equations, symmetries, dissipation laws and Newtonoid vector fields. We review the k-contact Euler-Lagrange equations written in terms of k-vector fields and sections and provide new results relating the solutions in both approaches. We also study different kind of symmetries depending on the structures they preserve: natural (preserving the Lagrangian function), dynamical (preserving the solutions), and k-contact (preserving the underlying geometric structures) symmetries. For some of these symmetries, we provide Noether-like theorems relating symmetries and dissipation laws. We also analyse the relation between k-contact symmetries and Newtonoid vector fields. Throughout the paper, we will use the damped vibrating string as our main illustrative example.

math-ph↗
arXiv2019

Tulczyjew's derivations and intrinsic field equations in classical field theories

This work presents the variational principles and the intrinsic versions of several equations in field theories, in particular, for the Classical Euler-Lagrange field equations, the implicit Euler-Lagrange field equations and the non-holonomic implicit Euler-Lagrange field equations. The advantages of the variational and intrinsic versions of these equations is that the Lagrangians functions are not necessary regular Lagrangians. We present two examples of this situation: Navier's equations and the non-holonomic Cosserat rod. Finally we comment the Hamiltonian case when the Lagrangian is a hyperregular function.

math-ph↗
arXiv2014

Symmetries in Lagrangian Field Theory

By generalizing the cosymplectic setting for time-dependent Lagrangian mechanics, we propose a geometric framework for the Lagrangian formulation of classical field theories with a Lagrangian depending on the independent variables. For that purpose we consider the first order jet bundles $J^1\pi$ of a fiber bundle $\pi:E\to {\mathbb R}^k$ where ${\mathbb R}^k$ is the space of independent variables. Generalized symmetries of the Lagrangian are introduced and the corresponding Noether Theorem is proved.

math-ph↗
arXiv2013

Reduction of polysymplectic manifolds

The aim of this paper is to generalize the classical Marsden-Weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which in- herit the polysymplectic structure. This generalization allows us to reduce polysymplectic Hamiltonian systems with symmetries, such as those appearing in certain kinds of classical field theories. As an application of this technique, an analogous to the Kirillov-Kostant-Souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed. Our procedure corrects some mistakes and inaccuracies in previous papers [29, 50] on this subject.

math-ph↗
arXiv2012

Symmetries, Newtonoids vector fields and conservation laws in the Lagrangian $k$-symplectic formalism

In this paper we study symmetries, Newtonoid vector fields, conservation laws, Noether's Theorem and its converse, in the framework of the $k$-symplectic formalism, using the Fr\"olicher-Nijenhuis formalism on the space of $k^1$-velocities of the configuration manifold. For the case $k=1$, it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether's Theorem and its converse. For the case $k>1$, we provide a new proof for Noether's Theorem, which shows that, in the $k$-symplectic formalism, each Cartan symmetry induces a conservation law. We prove that, under some assumptions, the converse of Noether's Theorem is also true and we provide examples when this is not the case. We also study the relations between dynamical symmetries, Newtonoid vector fields, Cartan symmetries and conservation laws, showing when one of them will imply the others. We use several examples of partial differential equations to illustrate when these concepts are related and when they are not.

math-ph↗
arXiv2008

Higher-order Cartan symmetries in k-symplectic field theory

For k-symplectic Hamiltonian field theories, we study infinitesimal transformations generated by certain kinds of vector fields which are not Noether symmetries, but which allow us to obtain conservation laws by means of a suitable generalization of the Noether theorem.

math-ph↗
arXiv2007

On the k-Symplectic, k-Cosymplectic and Multisymplectic Formalisms of Classical Field Theories

The objective of this work is twofold: First, we analyze the relation between the k-cosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms in classical field theories. In particular, we prove the equivalence between k-symplectic field theories and the so-called autonomous k-cosymplectic field theories, extending in this way the description of the symplectic formalism of autonomous systems as a particular case of the cosymplectic formalism in non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric character of the solutions to the Hamilton-de Donder-Weyl and the Euler-Lagrange equations in these formalisms. Second, we study the equivalence between k-cosymplectic and a particular kind of multisymplectic Hamiltonian and Lagrangian field theories (those where the configuration bundle of the theory is trivial).

math-ph↗
arXiv2007

Symmetries and conservation laws in the Gunther k-symplectic formalism of field theory

This paper is devoted to studying symmetries of k-symplectic Hamiltonian and Lagrangian first-order classical field theories. In particular, we define symmetries and Cartan symmetries and study the problem of associating conservation laws to these symmetries, stating and proving Noether's theorem in different situations for the Hamiltonian and Lagrangian cases. We also characterize equivalent Lagrangians, which lead to an introduction of Lagrangian gauge symmetries, as well as analyzing their relation with Cartan symmetries.

math-ph↗
arXiv2006

k-cosymplectic classical field theories: Tulczyjew, Skinner--Rusk and Lie-algebroid formulations

The k-cosymplectic Lagrangian and Hamiltonian formalisms of first-order field theories are reviewed and completed. In particular, they are stated for singular and almost-regular systems. Subsequently, several alternative formulations for k-cosymplectic first-order field theories are developed: First, generalizing the construction of Tulczyjew for mechanics, we give a new interpretation of the classical field equations in terms of certain submanifolds of the tangent bundle of the $k^1$-velocities of a manifold. Second, the Lagrangian and Hamiltonian formalisms are unified by giving an extension of the Skinner-Rusk formulation on classical mechanics. Finally, both formalisms are formulated in terms of Lie algebroids.

math-ph↗
arXiv2002

Geometric Structures in Field Theory

This review paper is concerned with the generalizations to field theory of the tangent and cotangent structures and bundles that play fundamental roles in the Lagrangian and Hamiltonian formulations of classical mechanics. The paper reviews, compares and constrasts the various generalizations in order to bring some unity to the field of study. The generalizations seem to fall into two categories. In one direction some have generalized the geometric structures of the bundles, arriving at the various axiomatic systems such as k-symplectic and k-tangent structures. The other direction was to fundamentally extend the bundles themselves and to then explore the natural geometry of the extensions. This latter direction gives us the multisymplectic geometry on jet and cojet bundles and n-symplectic geometry on frame bundles.

math-ph↗