41 research outputs found
On a Order Reduction Theorem in the Lagrangian Formalism
We provide a new proof of a important theorem in the Lagrangian formalism
about necessary and sufficient conditions for a second-order variational system
of equations to follow from a first-order Lagrangian.Comment: 9 pages, LATEX, no figures; appear in Il Nuovo Cimento
Homogeneous variational problems: a minicourse
A Finsler geometry may be understood as a homogeneous variational problem,
where the Finsler function is the Lagrangian. The extremals in Finsler geometry
are curves, but in more general variational problems we might consider extremal
submanifolds of dimension . In this minicourse we discuss these problems
from a geometric point of view.Comment: This paper is a written-up version of the major part of a minicourse
given at the sixth Bilateral Workshop on Differential Geometry and its
Applications, held in Ostrava in May 201
Linearization of nonlinear connections on vector and affine bundles, and some applications
A linear connection is associated to a nonlinear connection on a vector
bundle by a linearization procedure. Our definition is intrinsic in terms of
vector fields on the bundle. For a connection on an affine bundle our procedure
can be applied after homogenization and restriction. Several applications in
Classical Mechanics are provided
The Tulczyjew triple for classical fields
The geometrical structure known as the Tulczyjew triple has proved to be very
useful in describing mechanical systems, even those with singular Lagrangians
or subject to constraints. Starting from basic concepts of variational
calculus, we construct the Tulczyjew triple for first-order Field Theory. The
important feature of our approach is that we do not postulate {\it ad hoc} the
ingredients of the theory, but obtain them as unavoidable consequences of the
variational calculus. This picture of Field Theory is covariant and complete,
containing not only the Lagrangian formalism and Euler-Lagrange equations but
also the phase space, the phase dynamics and the Hamiltonian formalism. Since
the configuration space turns out to be an affine bundle, we have to use affine
geometry, in particular the notion of the affine duality. In our formulation,
the two maps and which constitute the Tulczyjew triple are
morphisms of double structures of affine-vector bundles. We discuss also the
Legendre transformation, i.e. the transition between the Lagrangian and the
Hamiltonian formulation of the first-order field theor
A Generalization of Chetaev's Principle for a Class of Higher Order Non-holonomic Constraints
The constraint distribution in non-holonomic mechanics has a double role. On
one hand, it is a kinematic constraint, that is, it is a restriction on the
motion itself. On the other hand, it is also a restriction on the allowed
variations when using D'Alembert's Principle to derive the equations of motion.
We will show that many systems of physical interest where D'Alembert's
Principle does not apply can be conveniently modeled within the general idea of
the Principle of Virtual Work by the introduction of both kinematic constraints
and variational constraints as being independent entities. This includes, for
example, elastic rolling bodies and pneumatic tires. Also, D'Alembert's
Principle and Chetaev's Principle fall into this scheme. We emphasize the
geometric point of view, avoiding the use of local coordinates, which is the
appropriate setting for dealing with questions of global nature, like
reduction.Comment: 27 pages. Journal of Mathematical Physics (to zappear
A setting for higher order differential equations fields and higher order Lagrange and Finsler spaces
We use the Fr\"olicher-Nijenhuis formalism to reformulate the inverse problem
of the calculus of variations for a system of differential equations of order
2k in terms of a semi-basic 1-form of order k. Within this general context, we
use the homogeneity proposed by Crampin and Saunders in [14] to formulate and
discuss the projective metrizability problem for higher order differential
equation fields. We provide necessary and sufficient conditions for higher
order projectivpre-e metrizability in terms of homogeneous semi-basic 1-forms.
Such a semi-basic 1-form is the Poincar\'e-Cartan 1-form of a higher order
Finsler function, while the potential of such semi-basic 1-form is a higher
order Finsler function.Comment: final, pre-published versio
Higher-order Mechanics: Variational Principles and other topics
After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the
Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we
state a unified geometrical version of the Variational Principles which allows
us to derive the Lagrangian and Hamiltonian equations for these kinds of
systems. Then, the standard Lagrangian and Hamiltonian formulations of these
principles and the corresponding dynamical equations are recovered from this
unified framework.Comment: New version of the paper "Variational principles for higher-order
dynamical systems", which was presented in the "III Iberoamerican Meeting on
Geometry, Mechanics and Control" (Salamanca, 2012). The title is changed. A
detailed review is added. Sections containing results about variational
principles are enlarged with additional comments, diagrams and summarizing
results. Bibliography is update
Symmetry aspects of nonholonomic field theories
The developments in this paper are concerned with nonholonomic field theories
in the presence of symmetries. Having previously treated the case of vertical
symmetries, we now deal with the case where the symmetry action can also have a
horizontal component. As a first step in this direction, we derive a new and
convenient form of the field equations of a nonholonomic field theory.
Nonholonomic symmetries are then introduced as symmetry generators whose
virtual work is zero along the constraint submanifold, and we show that for
every such symmetry, there exists a so-called momentum equation, describing the
evolution of the associated component of the momentum map. Keeping up with the
underlying geometric philosophy, a small modification of the derivation of the
momentum lemma allows us to treat also generalized nonholonomic symmetries,
which are vector fields along a projection. Such symmetries arise for example
in practical examples of nonholonomic field theories such as the Cosserat rod,
for which we recover both energy conservation (a previously known result), as
well as a modified conservation law associated with spatial translations.Comment: 18 page
A complete lift for semisprays
In this paper, we define a complete lift for semisprays. If is a
semispray on a manifold , its complete lift is a new semispray on
. The motivation for this lift is two-fold: First, geodesics for
correspond to the Jacobi fields for , and second, this complete lift
generalizes and unifies previously known complete lifts for Riemannian metrics,
affine connections, and regular Lagrangians. When is a spray, we prove that
the projective geometry of uniquely determines . We also study how
symmetries and constants of motions for lift into symmetries and constants
of motions for
Reduction of invariant constrained systems using anholonomic frames
We analyze two reduction methods for nonholonomic systems that are invariant
under the action of a Lie group on the configuration space. Our approach for
obtaining the reduced equations is entirely based on the observation that the
dynamics can be represented by a second-order differential equations vector
field and that in both cases the reduced dynamics can be described by
expressing that vector field in terms of an appropriately chosen anholonomic
frame.Comment: 19 page