1,036 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
Multivector Fields and Connections. Setting Lagrangian Equations in Field Theories
The integrability of multivector fields in a differentiable manifold is
studied. Then, given a jet bundle , it is shown that integrable
multivector fields in are equivalent to integrable connections in the
bundle (that is, integrable jet fields in ). This result is
applied to the particular case of multivector fields in the manifold and
connections in the bundle (that is, jet fields in the repeated jet
bundle ), in order to characterize integrable multivector fields and
connections whose integral manifolds are canonical lifting of sections. These
results allow us to set the Lagrangian evolution equations for first-order
classical field theories in three equivalent geometrical ways (in a form
similar to that in which the Lagrangian dynamical equations of non-autonomous
mechanical systems are usually given). Then, using multivector fields; we
discuss several aspects of these evolution equations (both for the regular and
singular cases); namely: the existence and non-uniqueness of solutions, the
integrability problem and Noether's theorem; giving insights into the
differences between mechanics and field theories.Comment: New sections on integrability of Multivector Fields and applications
to Field Theory (including some examples) are added. The title has been
slightly modified. To be published in J. Math. Phy
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
Lagrangian-Hamiltonian unified formalism for field theory
The Rusk-Skinner formalism was developed in order to give a geometrical
unified formalism for describing mechanical systems. It incorporates all the
characteristics of Lagrangian and Hamiltonian descriptions of these systems
(including dynamical equations and solutions, constraints, Legendre map,
evolution operators, equivalence, etc.).
In this work we extend this unified framework to first-order classical field
theories, and show how this description comprises the main features of the
Lagrangian and Hamiltonian formalisms, both for the regular and singular cases.
This formulation is a first step toward further applications in optimal control
theory for PDE's.Comment: LaTeX file, 23 pages. Minor changes have been made. References are
update
Extended Hamiltonian systems in multisymplectic field theories
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
non-autonomous 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.Comment: 36 pp. The introduction and the abstract have been rewritten. New
references are added and some little mistakes are corrected. The title has
been slightly modifie
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
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
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