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In this paper we study coisotropic reduction in different types of dynamics according to the geometry of the corresponding phase space. The relevance of 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. [ABSTRACT FROM AUTHOR]

In this paper, we study internal properties of Cosserat media. In fact, by using groupoids and smooth distributions, we obtain three canonical equations. The non-holonomic material equation for Cosserat media characterizes the uniformity of the material. The holonomic material equation for Cosserat media permits us to study when a Cosserat material is a second-grade material. It is remarkable that these two equations also provide us a unique and maximal division of the Cosserat medium into uniform and second-grade parts, respectively. Finally, we present a proper definition of homogeneity of the Cosserat medium, which does not need to assume uniformity. Thus, the homogeneity equation for Cosserat media characterizes this notion of homogeneity. [ABSTRACT FROM AUTHOR]

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 problem, which is simultaneously a generalization of the Herglotz variational principle and an optimal control problem. An application to the study of a thermodynamic system is provided. [ABSTRACT FROM AUTHOR]

For any body-time manifold R × B 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 to deal with general non-uniform material evolution. [ABSTRACT FROM AUTHOR]

Measuring the societal impact of research has become a key issue in scientific research worldwide. Most competitive research funding agencies demand societal impact in research proposals. We must not only bring science closer to the public, but also make it improve the condition in which the science is found. But how can we measure the societal impact of a research once it has been completed? Through this article we present the Social Impact ex-post Evaluation Protocol (SIEP), a proposal of a tool to gather and measure the social impact of research. It is based on the qualitative work developed in the framework of NET4IMPACT, a Research Network on the Social Impact of Science. The work was developed applying communicative methodology, comparing across disciplines, and culminating in the establishment of a tool that can help in the measurement of societal impact from a great diversity of scientific areas. [ABSTRACT FROM AUTHOR]

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. Furthermore, we consider the reduction of the Hamilton–Jacobi problem for a Čaplygin system to the Hamilton–Jacobi problem for a forced Lagrangian system. [ABSTRACT FROM AUTHOR]

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. This bracket was defined in [6,21] although there was already some particular and less direct definition. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids [13,20]. Recently, reviewing two older papers by R. J. Eden [17,18] , 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. [ABSTRACT FROM AUTHOR]

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 variational calculus with constraints on a Lagrangian with a dissipative variable. The variation of the energy and the other dissipative quantities is also obtained, giving the usual results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of mechanical systems subjected to external forces in the framework of symplectic geometry. We obtain 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., external forces that are derived from a dissipation function, and illustrate them with some examples. Moreover, we present a theory for the reduction in Lagrangian systems subjected to external forces, which are invariant under the action of a Lie group. [ABSTRACT FROM AUTHOR]

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 bundles, whose dynamic takes place in open subsets of the general manifold. We present a lcs Hamilton–Jacobi equation in subsets of the general manifold and then provide a global view by using the Lichnerowicz–deRham differential. We show a comparison between the global and local description of a lcs Hamilton–Jacobi theory, and how actually the local behavior can be glued to retrieve the global behavior of the Hamilton–Jacobi theory. A particular example is the case of Gaussian isokinetic dynamics in which we apply our structure in certain submanifolds of the phase space. [ABSTRACT FROM AUTHOR]

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 preservation of isolated systems and non-decreasing total entropy, respectively. We completely clarify its qualitative dynamics, the underlying geometrical structures and we also show how to apply discrete gradient methods to numerically integrate the evolution equations for these systems. [ABSTRACT FROM AUTHOR]

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 algorithm (for singular or regular Lagrangians), and is especially useful to describe singular Lagrangian systems. Some examples are also discussed to illustrate the method. A unified geometric framework is presented 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 algorithm (for singular or regular Lagrangians), and is especially useful to describe singular Lagrangian systems. Some examples are also discussed to illustrate the method. [ABSTRACT FROM AUTHOR]

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 subdivision of the material body into strictly smoothly uniform components. Thus, the constitutive law induces a unique partition of the body into smoothly uniform sub-bodies, laminates, filaments and isolated points. [ABSTRACT FROM AUTHOR]

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 construct the Hamiltonian counterpart and prove the equivalence with the Lagrangian side. A Dirac–Jacobi bracket is constructed similar to the Dirac bracket. [ABSTRACT FROM AUTHOR]

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 the dynamics of contact Hamiltonian systems. [ABSTRACT FROM AUTHOR]

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 the state of deformation of the material body, one can automatically associate to it a groupoid. Under conditions of differentiability, this material groupoid is a Lie groupoid. Its associated Lie algebroid plays an important role in the determination of the existence of material defects, such as dislocations. This paper presents a rather intuitive treatment of these ideas. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Associated to each material body B there exists a groupoid Ω B consisting of all the material isomorphisms connecting the points of B . The uniformity character of B is reflected in the properties of Ω B : B is uniform if, and only if, Ω B is transitive. Smooth uniformity corresponds to a Lie groupoid and, specifically, to a Lie subgroupoid of the groupoid Π 1 B , B of 1-jets of B . We consider a general situation when Ω B is only an algebraic subgroupoid. Even in this case, we can cover B by a material foliation whose leaves are transitive. The same happens with Ω B and the corresponding leaves generate transitive Lie groupoids (roughly speaking, the leaves covering B ). This result opens the possibility to study the homogeneity of general material bodies using geometric instruments. [ABSTRACT FROM AUTHOR]

In this paper we present an application of the groupoid theory to the study of relevant case of material evolution phenomena, the 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 morphogenesis equation , to calculate the material distributions associated to this phenomenon. [ABSTRACT FROM AUTHOR]

Reduction theory has played a major role in the study of Hamiltonian systems. Whilst 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 relies on approximations of a complete solution of the Hamilton–Jacobi equation. The natural question that we address in this paper is how these two topics (reduction and Hamilton–Jacobi theory) fit together. We obtain a reduction and reconstruction procedure for the Hamilton–Jacobi equation with symmetries, even in a generalized sense to be clarified below. Several applications and relations to other reduction of the Hamilton–Jacobi theory are shown in the last section of the paper. It is remarkable that as by-product we obtain a generalization of the Ge–Marsden reduction procedure [18] and the results in [17] . Quite surprisingly, the classical ansätze available in the literature to solve the Hamilton–Jacobi equation (see [2,19] ) are also particular instances of our framework. [ABSTRACT FROM AUTHOR]

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 Lagrangians, which generalizes the symplectic case. We also present some results on the inverse problem for extended contact systems. [ABSTRACT FROM AUTHOR]

In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces. [ABSTRACT FROM AUTHOR]

In this paper, we extend the geometric formalism of the Hamilton-Jacobi theory for time-dependent Mechanics to the case of classical field theories in the k-cosymplectic framework. [ABSTRACT FROM AUTHOR]

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 be exactly solved or classified. In particular, we will solve the example of a conformal scalar field using the geometric Hamilton–Jacobi theory that is explicitly proposed for conformal fields on a multisymplectic manifold. To complete the geometric approach to study field theories, we propose the Hamilton–Jacobi theory for conformal fields in a Cauchy data space, in which space and time are split separately and the dynamics is depicted in an infinite-dimensional manifold. [ABSTRACT FROM AUTHOR]

We develop a Hamilton-Jacobi theory for singular Lagrangian systems in the Skinner-Rusk formalism. Comparisons with the Hamilton-Jacobi problem in the Lagrangian and Hamiltonian settings are discussed. [ABSTRACT FROM AUTHOR]

A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric structure allows us to describe in a simplified way the mechanics of nonholonomic systems with both control and external forces. [ABSTRACT FROM AUTHOR]

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 system, and establish the relations between these two approaches. In the last section, some examples are discussed. [ABSTRACT FROM AUTHOR]

We consider Hamiltonian systems in first-order multisymplectic field theories. We review the properties of Hamiltonian systems in the so-called restricted multimomentum bundle, including the variational principle which leads to the Hamiltonian field equations. In an analogous way to how these systems are defined in the so-called extended (symplectic) formulation of nonautonomous mechanics, we introduce Hamiltonian systems in the extended multimomentum bundle. The geometric properties of these systems are studied, the Hamiltonian equations are analyzed using integrable multivector fields, the corresponding variational principle is also stated, and the relation between the extended and the restricted Hamiltonian systems is established. All these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories. [ABSTRACT FROM AUTHOR]

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach. [ABSTRACT FROM AUTHOR]

In this paper we develop the construction of geometric integrators for higher-order mechanics. These integrators are naturally symplectic and preserve momentum since they have a discrete variational origin. [ABSTRACT FROM AUTHOR]

We present a geometric algorithm for obtaining consistent solutions to systems of partial differential equations, mainly arising from singular covariant first-order classical field theories. This algorithm gives an intrinsic description of all the constraint submanifolds. The field equations are stated geometrically, either representing their solutions by integrable connections or, what is equivalent, by certain kinds of integrable m-vector fields. First, we consider the problem of finding connections or multivector fields solutions to the field equations in a general framework: a pre-multisymplectic fiber bundle (which will be identified with the first-order jet bundle and the multi-momentum bundle when Lagrangian and Hamiltonian field theories are considered). Then, the problem is stated and solved in a linear context, and a pointwise application of the results leads to the algorithm for the general case. In a second step, the integrability of the solutions is also studied. Finally, the method is applied to Lagrangian and Hamiltonian field theories and, for the former, the problem of finding holonomic solutions is also analyzed. [ABSTRACT FROM AUTHOR]

In this paper we study general symmetries for optimal control problems making use of the geometric formulation proposed in [7]. This framework allows us to reduce the number of equations associated with optimal control problems with symmetry and compare the solutions of the original system with the solutions of the reduced one. The reconstruction of the optimal controls starting from the reduced problem is also explored. [Copyright &y& Elsevier]

Lichnerowicz–Jacobi cohomology of Jacobi manifolds is reviewed. The use of the associated Lie algebroid allows to prove that the Lichnerowicz–Jacobi cohomology is invariant under conformal changes of the Jacobi structure. We also compute the Lichnerowicz–Jacobi cohomology for a large variety of examples. [Copyright &y& Elsevier]

A multisymplectic setting for classical field theories subject to nonholonomic constraints is presented. The infinite-dimensional setting in the space of Cauchy data is also given. [Copyright &y& Elsevier]

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, T k Q × R , whose geometric structures are previously introduced in order to state the Lagrangian and Hamiltonian formalisms for these kinds of systems, including their variational formulation. The variational principle, the contact forms, and the geometric dynamical equations are obtained by using those structures and generalizing the standard formulation of contact Lagrangian and Hamiltonian systems. As an alternative approach, we develop a unified description that encompasses the Lagrangian and Hamiltonian equations as well as their relationship through the Legendre map; all of them are obtained from the contact dynamical equations and the constraint algorithm that is implemented because, in this formalism, the dynamical systems are always singular. Some interesting examples are finally analysed using these geometric formulations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a constraint algorithm for time-dependent Lagrangian systems which permit us to solve the motion equations. This algorithm extends the Gotay and Nester algorithm for autonomous Lagrangians which is, in fact, a particular case. To do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used. [ABSTRACT FROM AUTHOR]

In this paper a description, a la Tulczyjew, of constrained time-dependent dynamical systems in terms of special symplectic manifolds is first developed. Next, this description is combined with the constraint algorithm for time-dependent constrained systems. [ABSTRACT FROM AUTHOR]

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 that dissipate at the same rate than the energy; so, the corresponding quotients are true conserved quantities. [ABSTRACT FROM AUTHOR]

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 analyze 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. [ABSTRACT FROM AUTHOR]

In this paper a new interpretation of the Hamiltonization of higher-order Lagrangian field theories is given, introducing a new class of tensor fields of type (1, 1) that extends almost-tangent structures to the tangent bundle of nk velocities. [ABSTRACT FROM AUTHOR]

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 dynamics. This fact is here amended by a generating family of Morse functions which plays the role of a Hamiltonian. A Hamilton–Jacobi equation is obtained with the aid of this generating family of functions. To conclude, we apply our results to singular Lagrangians by employing the construction of special symplectic structures. [ABSTRACT FROM AUTHOR]

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations. [ABSTRACT FROM AUTHOR]

We develop a Hamilton-Jacobi theory for singular lagrangian systems using the Gotay-Nester-Hinds constraint algorithm. The procedure works even if the system has secondary constraints. [ABSTRACT FROM AUTHOR]

A geometrical description of classical field theories of first order is given. The underlying k-cosymplectic structure permits to derive the corresponding field equations. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

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 a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew's triple is constructed for evolution contact dynamics. [ABSTRACT FROM AUTHOR]