36 research outputs found
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
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
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
On the k-Symplectic, k-Cosymplectic and Multisymplectic Formalisms of Classical Field Theories
The objective of this work is twofold: First, we analyze the relation between
the k-cosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms
in classical field theories. In particular, we prove the equivalence between
k-symplectic field theories and the so-called autonomous k-cosymplectic field
theories, extending in this way the description of the symplectic formalism of
autonomous systems as a particular case of the cosymplectic formalism in
non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric
character of the solutions to the Hamilton-de Donder-Weyl and the
Euler-Lagrange equations in these formalisms. Second, we study the equivalence
between k-cosymplectic and a particular kind of multisymplectic Hamiltonian and
Lagrangian field theories (those where the configuration bundle of the theory
is trivial).Comment: 25 page
Properties of Multisymplectic Manifolds
This lecture is devoted to review some of the main properties of
multisymplectic geometry. In particular, after reminding the standard
definition of multisymplectic manifold, we introduce its characteristic
submanifolds, the canonical models, and other relevant kinds of multisymplectic
manifolds, such as those where the existence of Darboux-type coordinates is
assured. The Hamiltonian structures that can be defined in these manifolds are
also studied, as well as other important properties, such as their invariant
forms and the characterization by automorphisms.Comment: 10 pp. Changes in Sections 5 and 7 (where brief guides to the proofs
of theorems have been added). Lecture given at the workshop on {\sl Classical
and Quantum Physics: Geometry, Dynamics and Control. (60 Years Alberto Ibort
Fest), Instituto de Ciencias Matem\'aticas (ICMAT)}, Madrid (Spain), 5--9
March 201
Invariant Forms and Automorphisms of Locally Homogeneous Multisymplectic Manifolds
It is shown that the geometry of locally homogeneous multisymplectic
manifolds (that is, smooth manifolds equipped with a closed nondegenerate form
of degree > 1, which is locally homogeneous of degree k with respect to a local
Euler field) is characterized by their automorphisms. Thus, locally homogeneous
multisymplectic manifolds extend the family of classical geometries possessing
a similar property: symplectic, volume and contact. The proof of the first
result relies on the characterization of invariant differential forms with
respect to the graded Lie algebra of infinitesimal automorphisms, and on the
study of the local properties of Hamiltonian vector fields on locally
multisymplectic manifolds. In particular it is proved that the group of
multisymplectic diffeomorphisms acts (strongly locally) transitively on the
manifold. It is also shown that the graded Lie algebra of infinitesimal
automorphisms of a locally homogeneous multisymplectic manifold characterizes
their multisymplectic diffeomorphisms.Comment: 25 p.; LaTeX file. The paper has been partially rewritten. Some
terminology has been changed. The proof of some theorems and lemmas have been
revised. The title and the abstract are slightly modified. An appendix is
added. The bibliography is update
Non-standard connections in classical mechanics
In the jet-bundle description of first-order classical field theories there
are some elements, such as the lagrangian energy and the construction of the
hamiltonian formalism, which require the prior choice of a connection. Bearing
these facts in mind, we analyze the situation in the jet-bundle description of
time-dependent classical mechanics. So we prove that this connection-dependence
also occurs in this case, although it is usually hidden by the use of the
``natural'' connection given by the trivial bundle structure of the phase
spaces in consideration. However, we also prove that this dependence is
dynamically irrelevant, except where the dynamical variation of the energy is
concerned. In addition, the relationship between first integrals and
connections is shown for a large enough class of lagrangians.Comment: 17 pages, Latex fil