On the Multimomentum Bundles and the Legendre Maps in Field Theories

We study the geometrical background of the Hamiltonian formalism of first-order Classical Field Theories. In particular, different proposals of multimomentum bundles existing in the usual literature (including their canonical structures) are analyzed and compared. The corresponding Legendre maps are introduced. As a consequence, the definition of regular and almost-regular Lagrangian systems is reviewed and extended from different but equivalent ways.Comment: LaTeX file, 19 pages. Replaced with the published version. Minor mistakes are correcte

Geometry of Lagrangian First-order Classical Field Theories

We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in consideration. First of all, we construct all the geometric structures associated with a first-order jet bundle and, using them, we develop the lagrangian formalism, defining the canonical forms associated with a lagrangian density and the density of lagrangian energy, obtaining the {\sl Euler-Lagrange equations} in two equivalent ways: as the result of a variational problem and developing the {\sl jet field formalism} (which is a formulation more similar to the case of mechanical systems). A statement and proof of Noether's theorem is also given, using the latter formalism. Finally, some classical examples are briefly studied.Comment: Latex file, 48 page

A geometrical analysis of the field equations in field theory

In this review paper we give a geometrical formulation of the field equations in the Lagrangian and Hamiltonian formalisms of classical field theories (of first order) in terms of multivector fields. This formulation enables us to discuss the existence and non-uniqueness of solutions, as well as their integrability.Comment: 14 pages. LaTeX file. This is a review paper based on previous works by the same author

Remarks on multisymplectic reduction

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. Some comments added in the section "Discussion and outlook" and in the Acknowledgments. New references are added. Minor mistakes are correcte

Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries

We state the intrinsic form of the Hamiltonian equations of first-order Classical Field theories in three equivalent geometrical ways: using multivector fields, jet fields and connections. Thus, these equations are given in a form similar to that in which the Hamiltonian equations of mechanics are usually given. Then, using multivector fields, we study several aspects of these equations, such as the existence and non-uniqueness of solutions, and the integrability problem. In particular, these problems are analyzed for the case of Hamiltonian systems defined in a submanifold of the multimomentum bundle. Furthermore, the existence of first integrals of these Hamiltonian equations is considered, and the relation between {\sl Cartan-Noether symmetries} and {\sl general symmetries} of the system is discussed. Noether's theorem is also stated in this context, both the ``classical'' version and its generalization to include higher-order Cartan-Noether symmetries. Finally, the equivalence between the Lagrangian and Hamiltonian formalisms is also discussed.Comment: Some minor mistakes are corrected. Bibliography is updated. To be published in J. Phys. A: Mathematical and Genera

Symplectic Cuts and Projection Quantization

The recently proposed projection quantization, which is a method to quantize particular subspaces of systems with known quantum theory, is shown to yield a genuine quantization in several cases. This may be inferred from exact results established within symplectic cutting.Comment: 12 pages, v2: additional examples and a new reference to related wor

A general construction of Poisson brackets on exact multisymplectic manifolds

In this note the long standing problem of the definition of a Poisson bracket in the framework of a multisymplectic formulation of classical field theory is solved. The new bracket operation can be applied to forms of arbitary degree. Relevant examples are discussed and important properties are stated with proofs sketched.Comment: 8 pages LaTeX, Talk delivered at the 34th Symp. on Math. Phys., Torun, Poland, June 200

Reconstrucción y adaptación de bastidor en la estructura de un cuadricielo para generar un "Tri-car"

The aim of this project is to put into practice all the knowledge acquired during the four years of study in the electromechanical automotive degree program at University of San Francisco. A thorough investigation of everything that falls within the area of construction, adaptation, restoration and assembly of systems, mechanisms and parts which takes a "Tri-car" was carried out. It has been analyzed to detail all the aspects involved in the creation of a frame that fits the tubular structure of a quad, which was put into practice and served for the optimal development of this project.El presente proyecto tiene como finalidad poner en práctica todos los conocimientos obtenidos en los cuatro años de estudio en la carrera de electromecánica automotriz en la Universidad San Francisco de Quito. Se efectuó una investigación minuciosa de todo lo que compete al área de construcción, adaptación, restauración y ensamblaje de los sistemas, mecanismos y piezas que lleva un “Tri-car”. Por lo cual se analizó detalladamente los aspectos concernientes a la generación de un bastidor adaptable a la estructura tubular de un cuadriciclo, lo cual se puso en práctica y fue utilizado para el desarrollo óptimo del proyecto