arXiv subjects

Xavier Gràcia

Publications and source records attributed to Xavier Gràcia.

17 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

On the Unilateral Approximation Condition with Linear Arrays in Near-Field NLoS Propagation

This paper investigates the validity of the unilateral approximation in near-field non-line-of-sight (NLoS) propagation for large-scale linear antenna arrays. Utilizing multiport communication theory, the study evaluates whether the electromagnetic interaction from the receiver back to the transmitter can be neglected. The analysis examines both discrete arrays with fixed spacing and continuous arrays with fixed aperture as the number of antennas increases. Results show the approximation is asymptotically satisfied on average for both cases. This provides a theoretical foundation for using correlated Rayleigh fading models in physically consistent next-generation wireless systems.

eess.SP↗

arXiv2026

Pairs of differential forms: a framework for precontact geometry

Precontact manifolds extend contact geometry by weakening the maximal non-integrability condition of the defining $1$-form. We clarify the geometric foundations of this structure by studying general pairs of a $1$-form and a $2$-form under mild regularity conditions. We characterize them through their class, analyse the role of distinguished vector fields, such as Reeb or Liouville fields, and study other associated geometrical objects. Precontact structures are then treated as the special case of pairs formed by a nowhere-vanishing $1$-form and its exterior derivative. We also define Hamiltonian dynamics on precontact manifolds. Several examples are presented to illustrate the theory.

math.DG↗

arXiv2026

Time-dependent metrics and connections

Time-dependent structures often appear in differential geometry, particularly in the study of non-autonomous differential equations on manifolds. One may study the geodesics associated with a time-dependent Riemannian metric by extremizing the corresponding energy functional, but also through the introduction of a more general concept of time-dependent covariant derivative operator. This relies on the examination of connections on the product manifold $\mathbb{R}\times M$. For these time-dependent covariant derivatives we explore the notions of parallel transport, geodesics and torsion. We also define the derivative of a one-parameter family of connections.

math.DG↗

arXiv2025

The evolution operator connecting the Lagrangian and Hamiltonian formalisms for contact systems

Some mechanical systems with dissipation can be described within the framework of the so-called contact mechanics: a modified form of the Euler-Lagrange equations stemming from Herglotz's variational principle, which admits a geometric formulation in terms of contact geometry. On the other hand, the study of singular Lagrangian systems and Dirac's theory of constraints can be enhanced by using the evolution operator $K$ that connects the Lagrangian and Hamiltonian formalisms. The main purpose of this paper is to transpose this evolution operator to the case of contact mechanics, and to study some of its main properties. In particular, we show that it provides a geometric description of the evolution equations and it relates the Hamiltonian and Lagrangian constraints. To illustrate the theory, we provide examples of singular contact systems based on modified versions of the simple pendulum and the Cawley Lagrangian.

math-ph↗

arXiv2024

Asymptotic Analysis of Near-Field Coupling in Massive MISO and Massive SIMO Systems

This paper studies the receiver to transmitter antenna coupling in near-field communications with massive arrays. Although most works in the literature consider that it is negligible and approximate it by zero, there is no rigorous analysis on its relevance for practical systems. In this work, we leverage multiport communication theory to obtain conditions for the aforementioned approximation to be valid in MISO and SIMO systems. These conditions are then particularized for arrays with fixed inter-element spacing and arrays with fixed size.

eess.SP↗

arXiv2023

On Darboux theorems for geometric structures induced by closed forms

This work reviews the classical Darboux theorem for symplectic, presymplectic, and cosymplectic manifolds (which are used to describe regular and singular mechanical systems), and certain cases of multisymplectic manifolds, and extends it in new ways to k-symplectic and k-cosymplectic manifolds (all these structures appear in the geometric formulation of first-order classical field theories). Moreover, we discuss the existence of Darboux theorems for classes of precosymplectic, k-presymplectic, k-precosymplectic, and premultisymplectic manifolds, which are the geometrical structures underlying some kinds of singular field theories. Approaches to Darboux theorems based on flat connections associated with geometric structures are given, while new results on polarisations for (k-)(pre)(co)symplectic structures arise.

math.DG↗

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.

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.

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.

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.

math-ph↗

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.

math-ph↗

arXiv2016

The wave equation for stiff strings and piano tuning

We study the wave equation for a string with stiffness. We solve the equation and provide a uniqueness theorem with suitable boundary conditions. For a pinned string we compute the spectrum, which is slightly inharmonic. Therefore, the widespread scale of 12 equal divisions of the just octave is not the best choice to tune instruments like the piano. Basing on the theory of dissonance, we provide a way to tune the piano in order to improve its consonance. A good solution is obtained by tuning a note and its fifth by minimizing their beats.

physics.class-ph↗

arXiv2016

Structural aspects of Hamilton-Jacobi theory

In our previous papers [11,13] we showed that the Hamilton-Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton-Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (`slicing vector fields') on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton-Jacobi theory, by considering special cases like fibred manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.

math-ph↗

arXiv2004

On some aspects of the geometry of differential equations in physics

In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular differential equations, and partial differential equations in field theories. The geometric structures underlying these systems are presented and commented. The main results concerning these structures are stated and discussed, as well as their influence on the study of the differential equations with which they are related. Furthermore, research to be developed in these areas is also commented.

math-ph↗