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394

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Introduction

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April 1990 - October 1990

February 1986 - present

October 1975 - January 1986

## Publications

Publications (394)

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...

In this paper we present an application of the groupoid theory to the study of relevant case of material evolution phenomena, the \textit{process of morphogenesis}. Our theory is inspired by Walter Noll's theories of continuous distributions and provides a unifying and very simple framework of these phenomena. We present the explicit equation, the...

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 constrain...

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 Lagran...

Cosymplectic geometry has been proven to be a very useful geometric background to describe time-dependent Hamiltonian dynamics. In this work, we address the globalization problem of locally cosymplectic Hamiltonian dynamics that failed to be globally defined. We investigate both the geometry of locally conformally cosymplectic (abbreviated as LCC)...

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 pres...

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 cyclic coordinates. We illustrate the applicability of the symmetry reduction...

This paper is devoted to discrete mechanical systems subject to external forces. We introduce a discrete version of systems with Rayleigh-type forces, obtain the equations of motion and characterize the equivalence for these systems. Additionally, we obtain a Noether’s theorem and other theorem characterizing the Lie subalgebra of symmetries of a f...

We present several results on the inverse problem and equivalent contact Lagrangian systems. These problems naturally lead to consider smooth transformations on the z variable (i.e., reparametrizations of the action). We present the extended contact Lagrangian systems to formalize this notion. With this structure we define horizontal equivalence of...

In this survey, we review the classical Hamilton Jacobi theory from a geometric point of view in different geometric backgrounds. We propose a Hamilton Jacobi equation for different geometric structures attending to one particular characterization: whether they fulfill the Jacobi and Leibniz identities simultaneously, or if at least they satisfy on...

For any body-time manifold [Formula: see text] there exists a groupoid, called the material groupoid, encoding all the material properties of the material evolution. A smooth distribution, the material distribution, is constructed to deal with the case in which the material groupoid is not a Lie groupoid. This new tool provides a unified framework...

The aim of this paper is to study the evolution of a material point of a body by itself, and not the body as a whole. To do this, we construct a groupoid encoding all the intrinsic properties of the material point and its characteristic foliations, which permits us to define the evolution equation. We also discuss phenomena like remodeling and agin...

This paper discusses symplectic and cosymplectic reduction for autonomous and non-autonomous simple hybrid forced mechanical systems, respectively. We give general conditions on whether it is possible to perform symmetry reduction for simple hybrid Hamiltonian and Lagrangian systems subject to non-conservative external forces, as well as time-depen...

We show that the contact dynamics obtained from the Herglotz variational principle can be described as a constrained nonholonomic or vakonomic ordinary Lagrangian system depending on a dissipative variable with an adequate choice of one constraint. As a consequence, we obtain the dynamics of contact nonholonomic and vakonomic systems as an ordinary...

This paper is devoted to discrete mechanical systems subject to external forces. We introduce a discrete version of systems with Rayleigh-type forces, obtain the equations of motion and characterize the equivalence for these systems. Additionally, we obtain a Noether's theorem and other theorem characterizing the Lie subalgebra of symmetries of a f...

We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and...

We present several results on the inverse problem and equivalent contactLagrangian systems. These problems naturally lead to consider smooth transformations on the z variable (i.e., reparametrizations of the action). We present the extended contact Lagrangian systems to formalize this notion. With this structure we define horizontal equivalence of...

In this paper we propose a Hamilton-Jacobi theory for implicit contact Hamiltonian systems in two different ways. One is the understanding of implicit contact Hamiltonian dynamics as a Legendrian submanifold of the tangent contact space, and another is as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space...

In this paper, we develop a Hamilton-Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton-Jacobi problem for forced Hamiltonian systems with symmetry. Furthermo...

We show that the contact dynamics obtained from the Herglotz variational principle can be described as a constrained nonholonomic or vakonomic ordinary Lagrangian system depending on a dissipative variable with an adequate choice of one constraint. As a consequence we obtain the dynamics of contact nonholonomic and vakonomic systems as ordinary var...

In this article we inspect the dynamics of classical field theories with a local conformal behavior. Our interest in the multisymplectic setting comes from its suitable description of field theories, and the conformal character has been added to account for field theories that are scale invariant, flat spaces, and because some conformal fields can...

The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian...

The aim of this paper is to study the evolution of a material point of a body by itself, and not the body as a whole. To do this, we construct a groupoid encoding all the intrinsic properties of the particle and its characteristic foliations, which permits us to define the evolution equation. We also discuss phenomena like remodeling and aging.

We are proposing Tulczyjew's triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and...

In this paper we study vakonomic dynamics on contact systems with nonlinear constraints. In order to obtain the dynamics, we consider a space of admisible paths, which are the ones tangent to a given submanifold. Then, we find the critical points of the Herglotz action on this space of paths. This dynamics can be also obtained through an extended L...

In this article, we continue the program started in [2] of exploring an important class of thermodynamic systems from a geometric point of view. The contents of this paper and the one already published in [2] provide a geometrical formulation, which tries to shed more light on the properties of thermodynamic systems without claiming to be a definit...

In this paper we study vakonomic dynamics on contact systems with nonlinear constraints. In order to obtain the dynamics, we consider a space of admisible paths, which are the ones tangent to a given submanifold. Then, we find the critical points of the Herglotz action on this space of paths. This dynamics can be also obtained through an extended L...

In this paper, we continue the construction of variational integrators adapted to contact geometry started in Vermeeren et al. (J Phys A 52(44):445206, 2019), in particular, we introduce a discrete Herglotz Principle and the corresponding discrete Herglotz Equations for a discrete Lagrangian in the contact setting. This allows us to develop conveni...

The aim of this paper is to develop a Hamilton--Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton-Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonia...

In this article, we provide a Hamilton–Jacobi formalism on locally conformally symplectic (lcs) manifolds. We are interested in the Hamilton–Jacobi as an alternative method for formulating the dynamics, while our interest in the locally conformal character will account for physical theories described by Hamiltonians defined on well-behaved line bun...

In this paper, we aim at addressing the globalization problem of Hamilton–DeDonder–Weyl equations on a local k-symplectic framework and we introduce the notion of locally conformal k-symplectic (l.c.k-s.) manifolds. This formalism describes the dynamical properties of physical systems that locally behave like multi-Hamiltonian systems. Here, we des...

This paper is devoted to the study of mechanical systems subjected to external forces in the framework of symplectic geometry. We obtain a Noether's theorem for Lagrangian systems with external forces, among other results regarding symmetries and conserved quantities. We particularize our results for the so-called Rayleigh dissipation, i.e., extern...

We present a complete theory of higher-order autonomous contact mechanics, which allows us to describe higher-order dynamical systems with dissipation. The essential tools for the theory are the extended higher-order tangent bundles, TkQ×R, whose geometric structures are previously introduced in order to state the Lagrangian and Hamiltonian formali...

In this article, we continue the program started in our previous article of exploring an important class of thermodynamic systems from a geometric point of view. In order to model the time evolution of systems verifying the two laws of thermodynamics, we show that the notion of evolution vector field is adequate to appropriately describe such syste...

Contact Hamiltonian dynamics is a subject that has still a short history, but with relevant applications in many areas: thermodynamics, cosmology, control theory, and neurogeometry, among others. In recent years there has been a great effort to study this type of dynamics both in theoretical aspects and in its potential applications in geometric me...

We present a complete theory of higher-order autonomous contact mechanics, which allows us to describe higher-order dynamical systems with dissipation. The essential tools for the theory are the extended higher-order tangent bundles, ${\rm T}^kQ\times{\mathbb R}$, whose geometric structures are previously introduced in order to state the Lagrangian...

By means of the Jacobi structure associated with a contact structure, we use the so-called evolution vector field to propose a new characterization of isolated thermodynamical systems with friction, a simple but important class of thermodynamical systems which naturally satisfy the first and second laws of thermodynamics, i.e. total energy preserva...

We prove that, under some natural conditions, Hamiltonian systems on a contact manifold $C$ can be split into a Reeb dynamics on an open subset of $C$ and a Liouville dynamics on a submanifold of $C$ of codimension 1. For the Reeb dynamics we find an invariant measure. Moreover, we show that, under certain completeness conditions, the existence of...

In this paper we combine two main topics in mechanics and optimal control theory: contact Hamiltonian systems and Pontryagin Maximum Principle. As an important result, among others, we develop a contact Pontryagin Maximum Principle that permits to deal with optimal control problems with dissipation. We also consider the Herglotz optimal control pro...

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 pioneering 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...

To any simple body with a given smooth constitutive equation, a groupoid, known as the material groupoid, can be associated naturally. When the body is non-uniform, however, the material groupoid is generally not differentiable. In such cases, the analysis can be based on a new differential geometric construct called the material distribution, to w...

Using contact geometry we give a new characterization of a simple but important class of thermodynamical systems which naturally satisfy the first law of thermodynamics (total energy preservation) and the second law (increase of entropy). We completely clarify its qualitative dynamics, the underlying geometrical structures and we show how to use di...

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...

In this paper, we continue the construction of variational integrators adapted to contact geometry started in \cite{VBS}, in particular, we introduce a discrete Herglotz Principle and the corresponding discrete Herglotz Equations for a discrete Lagrangian in the contact setting. This allows us to develop convenient numerical integrators for contact...

Recently, M. de León et al. (Campos et al., 2015) have developed a geometrical description of Hamilton–Jacobi theory for multisymplectic field theory. In our paper we analyse in the same spirit a special kind of field theories which are gauge field theories. The Hamilton- Jacobi theory for this kind of fields is shown.

In this paper, we extend the well-known Noether theorem for Lagrangian systems to contact Lagrangian systems. We introduce a classification of infinitesimal symmetries and obtain the corresponding dissipated quantities. We notice that in contact dynamics, the existence of infinitesimal symmetries does not produce conserved quantities, but functions...

In this article we inspect the dynamics of classical field theories with a local conformal behavior. Our interest in the multisymplectic setting comes from its suitable description of field theories, and the conformal character has been added to account for field theories that are scale invariant, flat spaces, and because some conformal fields can...

In this paper we aim at addressing the globalization problem of Hamilton-DeDonder-Weyl equations on a local $k$-symplectic framework and we introduce the notion of {\it locally conformal $k$-symplectic (l.c.k-s.) manifolds}. This formalism describes the dynamical properties of physical systems that locally behave like multi-Hamiltonian systems. Her...

In this article we develop a theory of contact systems with nonholonomic constraints. We obtain the dynamics from Herglotz's variational principle, by restricting the variations so that they satisfy the nonholonomic constraints. We prove that the nonholonomic dynamics can be obtained as a projection of the unconstrained Hamiltonian vector field. Fi...

In this article we provide a Hamilton-Jacobi formalism in locally conformally symplectic manifolds. Our interest in the Hamilton-Jacobi theory comes from the suitability of this theory as an integration method for dynamical systems, whilst our interest in the locally conformal character will account for physical theories described by Hamiltonians d...

In this paper, we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We show how the dissipative dynamics can be interpreted as a Legendrian submanifold, and also prove a coisotropic reduction theorem similar to the one in symplectic mechanics; as a consequence, we get a method to reduce...

In this paper, we extend the well-known Noether theorem for Lagrangian systems to contact Lagrangian systems. We introduce a classification of infinitesimal symmetries and obtain the corresponding dissipated quantities. We notice that in contact dynamics, the existence of infinitesimal symmetries does not produce conserved quantities, but functions...

We discuss, in all generality, the reduction of a Hamilton — Jacobi theory for systems subject to nonholonomic constraints and invariant under the action of a group of symmetries. We consider nonholonomic systems subject to both linear and nonlinear constraints and with different positioning of such constraints with respect to the symmetries.

In this paper, we discuss the singular Lagrangian systems on the framework of contact geometry. These systems exhibit a dissipative behavior in contrast with the symplectic scenario. We develop a constraint algorithm similar to the presymplectic one studied by Gotay and Nester (the geometrization of the well-known Dirac–Bergmann algorithm). We also...

In this paper we discuss singular Lagrangian systems on the framework of contact geometry. These systems exhibit a dissipative behavior in contrast with the symplectic scenario. We develop a constraint algorithm similar to the presymplectic one studied by Gotay and Nester (the geometrization of the well-known Dirac-Bergman algorithm). We also const...

Recently, M. de Le\'on el al. ([8]) have developed a geometrical description of Hamilton-Jacobi theory for multisymplectic field theory. In our paper we analyse in the same spirit a special kind of field theories which are gauge field theories. The Hamilton-Jacobi theory for this kind of fields is shown.

Walter Noll’s trailblazing constitutive theory of material defects in smoothly uniform bodies is recast in the language of Lie groupoids and their associated Lie algebroids. From this vantage point the theory is extended to non-uniform bodies by introducing the notion of singular material distributions and the physically cognate idea of graded unif...

The global formulation of the higher-order Poincaré-Cartan form for Lagrangian field theories in the Calculus of Variations is re-examined in terms of the theory of lifts of tensor fields on manifold to its higher-order prolongated jet bundles.

Lie groupoids and their associated algebroids arise naturally in the study of the constitutive properties of continuous media. Thus, Continuum Mechanics and Differential Geometry illuminate each other in a mutual entanglement of theory and applications. Given any material property, such as the elastic energy or an index of refraction, affected by t...

In this paper, we present a generalization of a Hamilton--Jacobi theory to higher order implicit differential equations. We propose two different backgrounds to deal with higher order implicit Lagrangian theories: the Ostrogradsky approach and the Schmidt transform, which convert a higher order Lagrangian into a first order one. The Ostrogradsky ap...

A groupoid $\Omega \left( \mathcal{B} \right)$ called material groupoid is naturally associated to any simple body $\mathcal{B}$ (see [MAREMDL,FGM2,MEPMDLSEG]). The material distribution is introduced due to the (possible) lack of differentiability of the material groupoid (see [MD,CHARDIST]). Thus, the inclusion of these new objects in the theory...

In this paper we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We prove a coisotropic reduction theorem similar to the one in symplectic mechanics.

Non-holonomic mechanical systems are deserving more and more interest in recent years due to its applications in engineering sciences as well as its geometric richness. In this paper we discuss the reduction of the Hamilton-Jacobi theory for systems with simmetries in its total generality, that is, we consider linear or nonlinear constraints. The r...

A Lie groupoid, called \textit{second-order non-holonomic material Lie groupoid}, is associated in a natural way to any Cosserat media. This groupoid is used to give a new definition of homogeneity which does not depend on a reference crystal. The corresponding Lie algebroid, called \textit{second-order non-holonomic material Lie algebroid}, is use...

Associated to each material body $\mathcal{B}$ there exists a groupoid $\Omega \left( \mathcal{B} \right)$ consisting of all the material isomorphisms connecting the points of $\mathcal{B}$. The uniformity character of $\mathcal{B}$ is reflected in the properties of $\Omega \left( \mathcal{B} \right)$: $\mathcal{B}$ is uniform if, and only if, $\Om...

The concept of material distribution is introduced as describing the geometric material structure of a general non-uniform body. Any smooth constitutive law is shown to give rise to a unique smooth integrable singular distribution. Ultimately, the material distribution and its associated singular foliation result in a rigorous and unique subdivisio...

In this paper we develop a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framewo...

In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of $TT^*Q$ generated by Morse families. The implicit character implies the nonexistence of a Hamiltonian function describing the...

In this paper we propose a geometric Hamilton--Jacobi theory on a Nambu--Jacobi manifold. The advantange of a geometric Hamilton--Jacobi theory is that if a Hamiltonian vector field $X_H$ can be projected into a configuration manifold by means of a one-form $dW$, then the integral curves of the projected vector field $X_H^{dW}$ can be transformed i...

In this paper, we review the discrete Hamilton--Jacobi theory from a geometric point of view. In the discrete realm, the usual geometric interpretation of the Hamilton--Jacobi theory in terms of vector fields is not straightforward. Here, we propose two alternative interpretations: one is the interpretation in terms of projective flows, the second...

In this paper, we apply the geometric Hamilton–Jacobi theory to obtain solutions of classical hamiltonian systems that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure plays a central role in the theory of time-dependent hamiltonians, whilst the second is here used to treat classical hamilto...

A Lie groupoid, called material Lie groupoid, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called material Lie algebroid, is used to characterize the homogeneity property of the material. We also relate these results with the previously ones in terms of $G$-structures.

In this paper, we apply the geometric Hamilton--Jacobi theory to obtain solutions of Hamiltonian systems in Classical Mechanics, that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure plays a central role in the theory of time-dependent Hamiltonians, whilst the second is here used to treat cl...

The Hamilton-Jacobi theory is a formulation of Classical Mechanics equivalent to other formulations as Newton's equations, Lagrangian or Hamiltonian Mechanics. It is particulary useful for the identification of conserved quantities of a mechanical system. The primordial observation of a geometric Hamilton-Jacobi equation is that if a Hamiltonian ve...

The differential-geometric underpinnings of a unified theory of material uniformity and evolution are exposed in terms of the language of groupoids subordinate to geometric distributions. Both the standard theory of material uniformity and the extended theory of functionally graded materials are included in the formulation as well as their temporal...

Reduction theory has played a major role in the study of Hamiltonian systems. On the other hand, the Hamilton-Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators rely on approximations of a complete solution...