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Action-dependent field theories are systems where the Lagrangian or Hamiltonian depends on new variables that encode the action. After a friendly introduction, we make a quick presentation of a new mathematical framework for action-dependent field theory: multicontact geometry. The formalism is illustrated in a variety of action-dependent Lagrangians, some of which are regular and other singular, that come from some well-known theories whose Lagrangians have been modified to incorporate action-dependent terms. Detailed computations are provided, including the constraint algorithm, in both the Lagrangian and Hamiltonian formalisms. They include the Klein-Gordon equation, the Bosonic string theory, Metric-affine gravity, Maxwell's electromagnetism, (2+1)-dimensional gravity and Chern-Simons equation, and the heat equation and Burgers' equation., Comment: 37 pages, plus appendix and references

We study the geometry underlying mechanics and its application to describe autonomous and nonautonomous conservative dynamical systems of different types; as well as dissipative dynamical systems. We use different geometric descriptions to study the main properties and characteristics of these systems; such as their Lagrangian, Hamiltonian and unified formalisms, their symmetries, the variational principles, and others. The study is done mainly for the regular case, although some comments and explanations about singular systems are also included., Comment: 237 pages. This is a draft version of a future book. Comments are welcome

A geometric approach to Sundman transformation defined by basic functions for systems of second-order differential equations is developed and the necessity of a change of the tangent structure by means of the function defining the Sundman transformation is shown. Among other applications of such theory we study the linearisability of a system of second-order differential equations and in particular the simplest case of a second-order differential equation. The theory is illustrated with several examples.

A new geometric framework is developed to describe non-conservative classical field theories, which is based on multisymplectic and contact geometries. Assuming certain additional conditions and using the forms that define this multicontact structure, as well as other geometric elements that are derived from them, we can introduce variational field equations in the multicontact manifolds. These equations are stated using different geometric tools; namely, sections, multivector fields and Ehresmann connections in fiber bundles. Then, this framework can be adapted to the jet bundle description of classical field theories and the field equations are stated both in the Lagrangian and the Hamiltonian formalisms, which are discussed in the regular and the singular cases., Comment: 44 pp

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., Comment: 28pp

A geometric approach to Sundman infinitesimal time-reparametrisation is given and some of its applications are used to illustrate the general theory. Special emphasis is put on geodesic motions and systems described by mechanical type Lagrangians. The Jacobi metric appears as a particular case of a Sundman transformation., Comment: To appear in Journal of Nonlinear Mathematical Physics

The work done by Isaac Newton more than three hundred years ago, continues being a path to increase our knowledge of Nature. To better understand all the ideas behind it, one of the finest ways is to generalize them to wider situations. In this report we make a review of one of these enlargements, the one that bears the mechanical systems from the elementary homogeneous three dimensional Euclidean space to the more abstract geometry of a Riemannian manifold., Comment: 29 pages

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 variational calculus with constraints on a Lagrangian with a dissipative variable. The variation of the energy and the other dissipative quantities are also obtained giving the usual results., Comment: 32 pp. New sections are added (3.4, 4.3, 5.5). Other minor changes. The bibliography is updated

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 Lagrangian, including Lagrange multiplier terms. This theory has important applications in optimal control theory for Herglotz control problems, in which the cost function is given implicitly, through an ODE, instead of by a definite integral. Indeed, these control problems can be considered as particular cases of vakonomic contact systems, and we can use the Lagrangian theory of contact systems in order to understand their symmetries and dynamics.

Being able to parse code-switched (CS) utterances, such as Spanish+English or Hindi+English, is essential to democratize task-oriented semantic parsing systems for certain locales. In this work, we focus on Spanglish (Spanish+English) and release a dataset, CSTOP, containing 5800 CS utterances alongside their semantic parses. We examine the CS generalizability of various Cross-lingual (XL) models and exhibit the advantage of pre-trained XL language models when data for only one language is present. As such, we focus on improving the pre-trained models for the case when only English corpus alongside either zero or a few CS training instances are available. We propose two data augmentation methods for the zero-shot and the few-shot settings: fine-tune using translate-and-align and augment using a generation model followed by match-and-filter. Combining the few-shot setting with the above improvements decreases the initial 30-point accuracy gap between the zero-shot and the full-data settings by two thirds.

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 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 analyzed using these geometric formulations., Comment: 39 pp. Minor changes. References updated

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.

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., Comment: arXiv admin note: text overlap with arXiv:1905.07354

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., Comment: 24 pp. This is the final version of the article. Some minor misprints are corrected. Some bibliographycal references are uploaded

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., Comment: 26 pp. Minor corrections. The bibliography has been updated

Hybrid systems are characterized by having an interaction between continuous dynamics and discrete events. The contribution of this paper is to provide hybrid systems with a novel geometric formulation so that controls can be added. Using this framework we describe some new global controllability tests for hybrid control systems exploiting the geometry and the topology of the set of jump points, where the instantaneous change of dynamics take place. Controllability is understood as the existence of a feasible trajectory for the system joining any two given points. As a result we describe examples where none of the continuous control systems are controllable, but the associated hybrid system is controllable because of the characteristics of the jump set., Comment: 27 pages, 5 figures

We consider a regular distribution $\mathcal{D}$ in a Riemannian manifold $(M,g)$. The Levi-Civita connection on $(M,g)$ together with the orthogonal projection allow to endow the space of sections of $\mathcal{D}$ with a natural covariant derivative, the intrinsic connection. Hence we have two different covariant derivatives for sections of $\mathcal{D}$, one directly with the connection in $(M,g)$ and the other one with this intrinsic connection. Their difference is the second fundamental form of $\mathcal{D}$ and we prove it is a significant tool to characterize the involutive and the totally geodesic distributions and to give a natural formulation of the equation of motion for mechanical systems with constraints. The two connections also give two different notions of curvature, curvature tensors and sectional curvatures, which are compared in this paper with the use of the second fundamental form., Comment: 23 pages

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find methods to derive superposition rules, constants of motion, and invariant tensor fields relative to the evolution of the multisymplectic Lie system. Our results are illustrated with examples occurring in physics, mathematics, and control theory., Comment: 33 pages

In this review paper we discuss the different interpretations of the concept of connection in a fiber bundle and in a jet bundle, and relate it with first and second-order systems of partial differential equations (PDE's) and multivector fields. As particular cases we analyze the concepts of linear connections and connections in a manifold., Comment: 21 pp. A new section (5) has been added. On it, new concepts, results and properties about connections in a manifold have been developed. arXiv admin note: text overlap with arXiv:dg-ga/9505004

The problem of reduction of multisymplectic manifolds by the action of Lie groups is stated and discussed, as a previous step to give a fully covariant scheme of reduction for classical field theories with symmetries., Comment: 9 pages. The definition of bracket of Hamiltonian forms has been corrected (and an acknowledgment to Prof. C. Blacker has been added in the "Acknowledgments")

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., Comment: 26 pages. Minor changes (some minor mistakes are corrected)

We study the dynamics of a rigid body in a central gravitational field modeled as a Hamiltonian system with continuous rotational symmetries following the geometrical framework of Wang et al. Novelties of our work are the use the Reduced Energy Momentum for the stability analysis and the treatment of axisymmetric bodies. We explicitly show the existence of new relative equilibria and study their stability and bifurcation patterns.