143 research outputs found
Reduction and Reconstruction Aspects of Second-Order Dynamical Systems with Symmetry
We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three parts and we show how the integral curves of the original system can be reconstructed from the reduced dynamics. An illustrative example confirms the results
A Lie algebroid framework for non-holonomic systems
In order to obtain a framework in which both non-holonomic mechanical systems
and non-holonomic mechanical systems with symmetry can be described, we
introduce in this paper the notion of a Lagrangian system on a subbundle of a
Lie algebroid.Comment: 18 page
Finsler geodesics of Lagrangian systems through Routh reduction
We make use of a symmetry reduction technique called Routh reduction to show that the solutions of the Euler-Lagrange equations of a strongly convex autonomous Lagrangian which lie on a specific energy level can be thought of as geodesics of an associated Finsler function
Second-order dynamical systems of Lagrangian type with dissipation
We give a coordinate-independent version of the smallest set of necessary and sufficient conditions for a given system of second-order ordinary differential equations to be of Lagrangian form with additional dissipative forces
A generalization of Szebehely's inverse problem of dynamics
The so-called inverse problem of dynamics is about constructing a potential
for a given family of curves. We observe that there is a more general way of
posing the problem by making use of ideas of another inverse problem, namely
the inverse problem of the calculus of variations. We critically review and
clarify different aspects of the current state of the art of the problem
(mainly restricted to the case of planar curves), and then develop our more
general approach.Comment: 21 pages, to appear in Rep. Math. Phy
A generalization of Szebehely's inverse problem of dynamics in dimension three
Extending a previous paper, we present a generalization in dimension 3 of the
traditional Szebehely-type inverse problem. In that traditional setting, the
data are curves determined as the intersection of two families of surfaces, and
the problem is to find a potential V such that the Lagrangian L = T - V, where
T is the standard Euclidean kinetic energy function, generates integral curves
which include the given family of curves. Our more general way of posing the
problem makes use of ideas of the inverse problem of the calculus of variations
and essentially consists of allowing more general kinetic energy functions,
with a metric which is still constant, but need not be the standard Euclidean
one. In developing our generalization, we review and clarify different aspects
of the existing literature on the problem and illustrate the relevance of the
newly introduced additional freedom with many examples.Comment: 23 pages, to appear in Rep. Math. Phy
On the generalized Helmholtz conditions for Lagrangian systems with dissipative forces
In two recent papers necessary and sufficient conditions for a given system
of second-order ordinary differential equations to be of Lagrangian form with
additional dissipative forces were derived. We point out that these conditions
are not independent and prove a stronger result accordingly.Comment: 10 pages, accepted for publ in Z. Angew. Math. Mech
Symmetry reduction, integrability and reconstruction in k-symplectic field theory
We investigate the reduction process of a k-symplectic field theory whose
Lagrangian is invariant under a symmetry group. We give explicit coordinate
expressions of the resulting reduced partial differential equations, the
so-called Lagrange-Poincare field equations. We discuss two issues about
reconstructing a solution from a given solution of the reduced equations. The
first one is an interpretation of the integrability conditions, in terms of the
curvatures of some connections. The second includes the introduction of the
concept of a k-connection to provide a reconstruction method. We show that an
invariant Lagrangian, under suitable regularity conditions, defines a
`mechanical' k-connection.Comment: 37 page
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