arXiv subjects

Rubén Izquierdo-López

Publications and source records attributed to Rubén Izquierdo-López.

9 recordsLinked to original sources

arXiv2026

From 2-covariant tensor fields to Poisson and Jacobi structures

Poisson and Jacobi structures play a fundamental role in the geometric description of many systems arising in classical mechanics. In most cases, the corresponding bivector field is induced by a non-degenerate 2-covariant tensor field. In this paper, we present a unified framework for constructing the associated brackets by studying the Poisson and Jacobi structures induced by these tensors. More specifically, under suitable assumptions on the tensor, we derive a formula for computing the Schouten--Nijenhuis bracket of the associated bivector field in terms of the curvature of a certain distribution and the exterior derivative of a differential form. This formula provides the obstruction to the existence of a Poisson or Jacobi structure. To illustrate the theory, we recover the classical brackets associated with symplectic, locally conformally symplectic, contact, cosymplectic, and cocontact geometries. Finally, we characterise the conditions under which fat bundles and almost cosymplectic structures of order $p$ determine a Jacobi bracket.

math.DG↗

arXiv2026

The Spencer cohomology and integrability of multisymplectic structures

We study the integrability problem of multisymplectic structures, by identifying them as $G$-structures. Applying the theory of Spencer cohomology, we give conditions on a multisymplectic form for it to admit a chart in which it has constant coefficients. This general study allows for a rough classification of multisymplectic structures of constant linear type, depending on the natural action of the stabilizer group. The theory is illustrated by providing a scheme for proving a Darboux theorem, which is exemplified with several relevant cases. We also build linear types of multisymplectic forms $\varpi_j$ whose flatness strictly requires a condition of order $j$. Finally, the corresponding Lie algebras are computed in the case of field theories.

math.DG↗

arXiv2026

Regularization of singular time-dependent Lagrangian systems

One approach to studying the dynamics of a singular Lagrangian system is to attempt to regularize it, that is, to find an equivalent and regular system. In the case of time-independent singular Lagrangians, an approach due to \textit{A. Ibort} and \textit{J. Marín-Solano} is to use the coisotropic embedding theorem proved by \textit{M.J. Gotay} which states that any pre-symplectic manifold can be coisotropically embedded in a symplectic manifold. In this paper, we revisit these results and provide an alternative approach, also based on the coisotropic embedding theorem, that employs the Tulczyjew isomorphism and almost product structures, and allows for a slight generalization of the construction. In this revision, we also prove uniqueness of the Lagrangian regularization to first order. Furthermore, we extend our methodology to the case of time-dependent singular Lagrangians.

math.DG↗

arXiv2026

Brackets in multicontact geometry and multisymplectization

In this paper we introduce a graded bracket of forms on multicontact manifolds. This bracket satisfies a graded Jacobi identity as well as two different versions of the Leibniz rule, one of them being a weak Leibniz rule, extending the well-known notions in contact geometry. In addition, we develop the multisymplectization of multicontact structures to relate these brackets to the ones present in multisymplectic geometry and obtain the field equations in an abstract context. The Jacobi bracket also permits to study the evolution of observables and study the dissipation phenomena, which we also address. Finally, we apply the results to classical dissipative field theories.

math.DG↗

arXiv2025

A zoo of coisotropic embeddings

The aim of this paper is to extend the coisotropic embedding theorem obtained by M. J. Gotay for pre-symplectic manifolds to more general geometric settings: cosymplectic, contact, cocontact, $k$-symplectic, $k$-cosymplectic, $k$-contact, and multisymplectic manifolds. The results are obtained by applying a generic methodology, which gives more relevance to the potential applications. In that sense, this paper gives the fundamental basis to be able to apply the results to the so-called regularization problem of singular Lagrangian systems, both in mechanics and in classical field theories.

math.DG↗

arXiv2025

A description of classical field equations using extensions of graded Poisson brackets

As it is well-known, Poisson brackets play a fundamental role both in mechanics and in classical field theories. In this paper we develop a theory of extensions of graded Poisson brackets in graded Dirac manifolds. We then show how these extensions can be used to obtain the field equations of a particular theory as well as the evolution of forms of arbitrary order, in a similar way that ordinary Poisson brackets provide in mechanics.

math-ph↗

arXiv2025

Graded Poisson and Graded Dirac structures

There have been several attempts in recent years to extend the notions of symplectic and Poisson structures in order to create a suitable geometrical framework for classical field theories, trying to achieve a success similar to the use of these concepts in Hamiltonian mechanics. These notions always have a graded character, since the multisymplectic forms are of a higher degree than two. Another line of work has been to extend the concept of Dirac structures to these new scenarios. In the present paper we review all these notions, relate them and propose and study a generalization that (under some mild regularity conditions) includes them and is of graded nature. We expect this generalization to allow us to advance in the study of classical field theories, their integrability, reduction, numerical approximations and even their quantization.

math-ph↗

arXiv2024

Coisotropic reduction in Multisymplectic Geometry

In this paper we study coisotropic reduction in multisymplectic geometry. On the one hand, we give an interpretation of Hamiltonian multivector fields as Lagrangian submanifolds and prove that $k$-coisotropic submanifolds induce a Lie subalgebra in the algebra of Hamiltonian $(k-1)$-forms, similar to how coisotropic submanifolds in symplectic geometry induce a Lie subalgebra under the Poisson bracket. On the other hand, we extend the classical result of symplectic geometry of projection of Lagrangian submanifolds in coisotropic reduction to bundles of forms, which naturally carry a multisymplectic structure.

math.SG↗

arXiv2024

A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact Hamiltonian systems

In this paper we study the coisotropic reduction in different types of dynamics according to the geometry of the corresponding phase space. The relevance of the coisotropic reduction is motivated by the fact that these dynamics can always be interpreted as Lagrangian or Legendrian submanifolds. Furthermore, Lagrangian or Legendrian submanifolds can be reduced by a coisotropic one.

math.SG↗