31 research outputs found
On some aspects of the geometry of non integrable distributions and applications
We consider a regular distribution in a Riemannian manifold
. The Levi-Civita connection on together with the orthogonal
projection allow to endow the space of sections of with a natural
covariant derivative, the intrinsic connection. Hence we have two different
covariant derivatives for sections of , one directly with the
connection in and the other one with this intrinsic connection. Their
difference is the second fundamental form of 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 page
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
Sundman transformation and alternative tangent structures
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
Optimal control, contact dynamics and Herglotz variational problem
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
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