107 research outputs found
A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold
We review properties of so-called special conformal Killing tensors on a
Riemannian manifold and the way they give rise to a Poisson-Nijenhuis
structure on the tangent bundle . We then address the question of
generalizing this concept to a Finsler space, where the metric tensor field
comes from a regular Lagrangian function , homogeneous of degree two in the
fibre coordinates on . It is shown that when a symmetric type (1,1) tensor
field along the tangent bundle projection satisfies a
differential condition which is similar to the defining relation of special
conformal Killing tensors, there exists a direct recursive scheme again for
first integrals of the geodesic spray. Involutivity of such integrals,
unfortunately, remains an open problem.Comment: This is a contribution to the Proc. of workshop on Geometric Aspects
of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Comment on 'Conservation laws of higher order nonlinear PDEs and the variational conservation laws in the class with mixed derivatives'
In a recent paper (R Narain and A H Kara 2010 J. Phys. A: Math. Theor. 43 085205), the authors claim to be applying Noether's theorem to higher-order partial differential equations and state that in a large class of examples 'the resultant conserved flows display some previously unknown interesting 'divergence properties' owing to the presence of the mixed derivatives' (citation from their abstract). It turns out that what this obscure sentence is meant to say is that the vector whose divergence must be zero (according to Noether's theorem), turns out to have non-zero divergence and subsequently must be modified to obtain a true conservation law. Clearly this cannot be right: we explain in detail the main source of the error
Alternative kinetic energy metrics for Lagrangian systems
We examine Lagrangian systems on R-n with standard kinetic energy terms for the possibility of additional, alternative Lagrangians with kinetic energy metrics different to the Euclidean one. Using the techniques of the inverse problem in the calculus of variations we find necessary and sufficient conditions for the existence of such Lagrangians. We illustrate the problem in two and three dimensions with quadratic and cubic potentials. As an aside we show that the well-known anomalous Lagrangians for the Coulomb problem can be removed by switching on a magnetic field, providing an appealing resolution of the ambiguous quantizations of the hydrogen atom
Lifting geometric objects to the dual of the first jet bundle of a bundle fibred over R
We study natural lifting operations from a bundle E over R to the dual bundle
of its first-jet bundle. The main purpose is to define a complete lift of a
type (1,1) tensor field on E and to understand all features of its
construction. Various other lifting operations of tensorial objects on E are
needed for that purpose. We prove that the complete lift of a type (1,1) tensor
with vanishing Nijenhuis torsion gives rise to a Poisson-Nijenhuis structure on
the dual of the first-jet bundle, and discuss in detail how the construction of
associated Darboux-Nijenhuis coordinates can be carried out
Generalized submersiveness of second-order ordinary differential equations
We generalize the notion of submersive second-order differential equations by relaxing the condition that the decoupling stems from the tangent lift of a basic distribution. It is shown that this leads to adapted coordinates in which a number of first-order equations decouple from the remaining second-order ones
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