184 research outputs found
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
Geometric Hamilton-Jacobi Theory
The Hamilton-Jacobi problem is revisited bearing in mind the consequences
arising from a possible bi-Hamiltonian structure. The problem is formulated on
the tangent bundle for Lagrangian systems in order to avoid the bias of the
existence of a natural symplectic structure on the cotangent bundle. First it
is developed for systems described by regular Lagrangians and then extended to
systems described by singular Lagrangians with no secondary constraints. We
also consider the example of the free relativistic particle, the rigid body and
the electron-monopole system.Comment: 40 page
Structural aspects of Hamilton-Jacobi theory
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
Geometric Hamilton-Jacobi Theory for Nonholonomic Dynamical Systems
The geometric formulation of Hamilton--Jacobi theory for systems with
nonholonomic constraints is developed, following the ideas of the authors in
previous papers. The relation between the solutions of the Hamilton--Jacobi
problem with the symplectic structure defined from the Lagrangian function and
the constraints is studied. The concept of complete solutions and their
relationship with constants of motion, are also studied in detail. Local
expressions using quasivelocities are provided. As an example, the nonholonomic
free particle is considered.Comment: 22 p
Infinitesimal Time Reparametrisation and Its Applications
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. © 2022, The Author(s)
Structural aspects of HamiltonâJacobi theory
The final publication is available at Springer via http://dx.doi.org/10.1142/S0219887816500171In our previous papers [11, 13] we showed that the HamiltonâJacobi 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 HamiltonâJacobi 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 (âslicing vector fieldsâ) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamiltonâ 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.Peer ReviewedPostprint (author's final draft
Pre-multisymplectic constraint algorithm for field theories
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 fibre bundle (which will be identified with
the first-order jet bundle and the multimomentum 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
analized.Comment: 30 pp. Presented in the International Workshop on Geometric Methods
in Modern Physics (Firenze, April 2005
On some aspects of the geometry of differential equations in physics
In this review paper, we consider three kinds of systems of differential
equations, which are relevant in physics, control theory and other applications
in engineering and applied mathematics; namely: Hamilton equations, singular
differential equations, and partial differential equations in field theories.
The geometric structures underlying these systems are presented and commented.
The main results concerning these structures are stated and discussed, as well
as their influence on the study of the differential equations with which they
are related. Furthermore, research to be developed in these areas is also
commented.Comment: 21 page
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