arXiv subjects

Asier López-Gordón

Publications and source records attributed to Asier López-Gordón.

At least 19 recordsLinked to original sources

arXiv2026

Egorov-Type Semiclassical Limits for Open Quantum Systems with a Bi-Lindblad Structure

This paper develops a bridge between bi-Hamiltonian structures of Poisson--Lie type, contact Hamiltonian dynamics, and the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) formalism for open quantum systems. On the classical side, we consider bi-Hamiltonian systems defined by a Poisson pencil with non-trivial invariants. Using an exact symplectic realization, these invariants are lifted and projected onto a contact manifold, yielding a completely integrable contact Hamiltonian system and a Jacobi-commutative algebra of observables. On the quantum side, we introduce a class of contact-compatible Lindblad generators: GKSL evolutions whose dissipative part preserves a commutative $C^\ast$-subalgebra generated by the quantizations of the classical dissipated quantities, and whose Hamiltonian part admits an Egorov-type semiclassical limit to the contact dynamics. This construction provides a mathematical mechanism compatible with the semiclassical limit for pure dephasing, compatible with integrability and contact dissipation. An explicit Poisson--Lie pencil, inspired by deformed Euler top models, is developed as a fully worked-out example illustrating the resulting bi-Lindblad structure and its semiclassical behavior.

math-ph↗

arXiv2026

Homogeneity actions, N-manifolds, and the Frobenius theorem

This paper is devoted to $\mathbb{N}$-graded supermanifolds $\mathcal{M}$ whose grading is induced by a homogeneity action, i.e., a smooth action $\mathbb{R}\ni t\mapsto h_t$ of the multiplicative monoid of real numbers on $\mathcal{M}$. We show that the map $h_0$ is a smooth retraction onto a submanifold $M=h_0(\mathcal{M})$, and that $h_0:\mathcal{M}\to M$ is a fiber bundle with typical fiber $\mathbb{R}^{m|n}$. Using this homogeneity approach, we obtain a simple proof of a homogeneous version of the Frobenius theorem. If, in addition, $h_{-1}$ acts as the parity operator on $\mathcal{M}$, we provide a geometric characterization equivalent to the recent definition of $\mathbb{N}$-manifolds due to Bursztyn, Cueca, and Mehta in terms of sheaves of graded algebras with prescribed local models. As a consequence, the homogeneous Frobenius theorem for $\mathbb{N}$-manifolds proved by these authors appears as a special case of our more general result.

math.DG↗

arXiv2026

A Darboux classification of homogeneous Pfaffian forms on graded manifolds

We study the local classification problem for differential Pfaffian forms on a supermanifold $M$ that are homogeneous with respect to a given homogeneity structure on $M$. The most familiar examples of homogeneity structures are those associated with vector bundle structures. Our aim is to show that, for a homogeneous form of fixed degree, there exist homogeneous Darboux coordinates. As a consequence, we obtain Darboux-type normal forms for homogeneous Pfaffian forms, recovering as special cases the classical Darboux theorem together with its contact and presymplectic counterparts. To formulate an analogue of Darboux classification in the supergeometric setting, we associate to a differential form $α$ the characteristic distribution $χ(α)=\ker(α)\cap\ker(\mathrm{d}α)$, and define the class of $α$ as the rank of $χ(α)$. We prove that, under suitable regularity and constant-rank assumptions, this distribution completely determines the local equivalence problem for homogeneous Pfaffian forms. Our results apply equally well to ordinary (purely even) manifolds.

math.DG↗

arXiv2026

Geometric integrators for adiabatically closed simple thermodynamic systems

A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples.

math-ph↗

arXiv2025

Homogeneous bi-Hamiltonian structures and integrable contact systems

Bi-Hamiltonian structures can be utilised to compute a maximal set of functions in involution for certain integrable systems, given by the eigenvalues of the recursion operator relating both Poisson structures. We show that the recursion operator relating two compatible Jacobi structures cannot produce a maximal set of functions in involution. However, as we illustrate with an example, bi-Hamiltonian structures can still be used to obtain a maximal set of functions in involution on a contact manifold, at the cost of symplectisation.

math-ph↗

arXiv2025

Generalized hybrid momentum maps and reduction by symmetries of forced mechanical systems with inelastic collisions

This paper discusses reduction by symmetries for autonomous and non-autonomous forced mechanical systems with inelastic collisions. In particular, we introduce the notion of generalized hybrid momentum map and hybrid constants of the motion to give general conditions on whether it is possible to perform symmetry reduction for Hamiltonian and Lagrangian systems subject to non-conservative external forces and non-elastic impacts, as well as its extension to time-dependent mechanical systems subject to time-dependent external forces and time-dependent inelastic collisions. We illustrate the applicability of the method with examples and numerical simulations.

eess.SY↗

arXiv2025

Liouville-Arnold theorem for homogeneous symplectic and contact Hamiltonian systems

A Hamiltonian system is completely integrable (in the sense of Liouville) if there exist as many independent integrals of motion in involution as the dimension of the configuration space. Under certain regularity conditions, Liouville-Arnold theorem states that the invariant geometric structure associated with Liouville integrability is a fibration by Lagrangian tori (or, more generally, Abelian Lie groups), on which the motion is linear. In this paper, a Liouville-Arnold theorem for contact Hamiltonian systems is proven. More specifically, it is shown that, given a $(2n+1)$-dimensional completely integrable contact system, one can construct a foliation by $(n+1)$-dimensional Abelian Lie groups and induce action-angle coordinates in which the equations of motion are linearized. One important novelty with respect to previous attempts is that the foliation consists of $(n+1)$-dimensional coisotropic submanifolds given by the preimages of rays by the functions in involution. In order to prove the theorem, we first develop a version of Liouville-Arnold theorem for homogeneous functions on exact symplectic manifolds (which is of independent interest), and then apply the symplectization to obtain the contact case.

math.SG↗

arXiv2024

The geometry of dissipation

Dissipative phenomena manifest in multiple mechanical systems. In this dissertation, different geometric frameworks for modelling non-conservative dynamics are considered. The objective is to generalize several results from conservative systems to dissipative systems, specially those concerning the symmetries and integrability of these systems. More specifically, three classes of geometric frameworks modelling dissipative systems are considered: systems with external forces, contact systems and systems with impacts. The first two allow modelling a continuous dissipation of energy over time, while the latter also permits considering abrupt changes of energy in the instants of the impacts.

math-ph↗

arXiv2024

Hamilton-Jacobi theory for nonholonomic and forced hybrid mechanical systems

A hybrid system is a system whose dynamics is given by a mixture of both continuous and discrete transitions. In particular, these systems can be utilised to describe the dynamics of a mechanical system with impacts. Based on the approach by Clark, we develop a geometric Hamilton-Jacobi theory for forced and nonholonomic hybrid dynamical systems. We state the corresponding Hamilton-Jacobi equations for these classes of systems and apply our results to analyze some examples.

math-ph↗

arXiv2024

A new perspective on nonholonomic brackets and Hamilton-Jacobi theory

The nonholonomic dynamics can be described by the so-called nonholonomic bracket on the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket on the cotangent bundle of the configuration manifold. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids. Recently, reviewing two older papers by R. J. Eden, we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton-Jacobi theory and the quantization of nonholonomic systems.

math-ph↗

arXiv2023

On the integrability of hybrid Hamiltonian systems

A hybrid system is a system whose dynamics are controlled by a mixture of both continuous and discrete transitions. The integrability of Hamiltonian systems is often identified with complete integrability or Liouville integrability, that is, the existence of as many independent integrals of motion in involution as the dimension of the phase space. Under certain regularity conditions, Liouville-Arnold theorem states that the invariant geometric structure associated with Liouville integrability is a fibration by Lagrangian tori, on which motions are linear. In this paper, we study an extension of the Liouville-Arnold theorem for hybrid systems whose continuous dynamics is given by a Hamiltonian vector field and we state conditions on the impact map and the switching surface under which a hybrid Hamiltonian system is, in a certain sense, completely integrable.

math-ph↗

arXiv2023

Symmetries, conservation and dissipation in time-dependent contact systems

In contact Hamiltonian systems, the so-called dissipated quantities are akin to conserved quantities in classical Hamiltonian systems. In this paper, we prove a Noether's theorem for non-autonomous contact Hamiltonian systems, characterizing a class of symmetries which are in bijection with dissipated quantities. We also study other classes of symmetries which preserve (up to a conformal factor) additional structures, such as the contact form or the Hamiltonian function. Furthermore, making use of the geometric structures of the extended tangent bundle, we introduce additional classes of symmetries for time-dependent contact Lagrangian systems. Our results are illustrated with several examples. In particular, we present the two-body problem with time-dependent friction, which could be interesting in celestial mechanics.

math-ph↗

arXiv2023

Nonholonomic brackets: Eden revisited

The nonholonomic dynamics can be described by the so-called nonholonomic bracket in the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket in the cotangent bundle of the configuration manifold. This bracket was defined by Cantrijn et al. and Ibort et al., although there was already some particular and less direct definition. On the other hand, another bracket, also called noholonomic, was defined using the description of the problem in terms of almost Lie algebroids. Recently, reviewing two older papers by R. J. Eden, we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton-Jacobi theory and the quantization of nonholonomic systems.

math-ph↗

arXiv2023

Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems

In this paper, we study the integrability of contact Hamiltonian systems, both time-dependent and independent. In order to do so, we construct a Hamilton--Jacobi theory for these systems following two approaches, obtaining two different Hamilton--Jacobi equations. Compared to conservative Hamiltonian systems, contact Hamiltonian systems depend of one additional parameter. The fact of obtaining two equations reflects whether we are looking for solutions depending on this additional parameter or not. In order to illustrate the theory developed in this paper, we study three examples: the free particle with a linear external force, the freely falling particle with linear dissipation and the damped and forced harmonic oscillator.

math-ph↗

arXiv2023

Contact Lagrangian systems subject to impulsive constraints

We describe geometrically contact Lagrangian systems under impulsive forces and constraints, as well as instantaneous nonholonomic constraints which are not uniform along the configuration space. In both situations, the vector field describing the dynamics of a contact Lagrangian system is determined by defining projectors to evaluate the constraints by using a Riemannian metric. In particular, we introduce the Herglotz equations for contact Lagrangian systems subject to instantaneous nonholonomic constraints. Moreover, we provide a Carnot-type theorem for contact Lagrangian systems subject to impulsive forces and constraints, which characterizes the changes of energy due to contact-type dissipation and impulsive forces. We illustrate the applicability of the method with practical examples, in particular, a rolling cylinder on a springily surface and a rolling sphere on a non-uniform surface, both with dissipation.

math-ph↗

arXiv2022

Discrete Mechanics and Optimal Control for Passive Walking with Foot Slippage

Forced variational integrators are given by the discretization of the Lagrange-d'Alembert principle for systems subject to external forces, and have proved useful for numerical simulation studies of complex dynamical systems. In this paper we model a passive walker with foot slip by using techniques of geometric mechanics, and we construct forced variational integrators for the system. Moreover, we present a methodology for generating (locally) optimal control policies for simple hybrid holonomically constrained forced Lagrangian systems, based on discrete mechanics, applied to a controlled walker with foot slip in a trajectory tracking problem.

eess.SY↗

arXiv2022

Nonsmooth Herglotz variational principle

In this paper, the theory of smooth action-dependent Lagrangian mechanics (also known as contact Lagrangians) is extended to a non-smooth context appropriate for collision problems. In particular, we develop a Herglotz variational principle for non-smooth action-dependent Lagrangians which leads to the preservation of energy and momentum at impacts. By defining appropriately a Legendre transform, we can obtain the Hamilton equations of motion for the corresponding non-smooth Hamiltonian system. We apply the result to a billiard problem in the presence of dissipation.

math.OC↗

arXiv2022

Hybrid Routhian reduction for simple hybrid forced Lagrangian systems

This paper discusses Routh reduction for simple hybrid forced mechanical systems. We give general conditions on whether it is possible to perform symmetry reduction for a simple hybrid Lagrangian system subject to non-conservative external forces, emphasizing the case of case of cyclic coordinates. We illustrate the applicability of the symmetry reduction procedure with an example and numerical simulations.

math.DS↗