141 research outputs found
Driven cofactor systems and Hamilton-Jacobi separability
This is a continuation of the work initiated in a previous paper on so-called
driven cofactor systems, which are partially decoupling second-order
differential equations of a special kind. The main purpose in that paper was to
obtain an intrinsic, geometrical characterization of such systems, and to
explain the basic underlying concepts in a brief note. In the present paper we
address the more intricate part of the theory. It involves in the first place
understanding all details of an algorithmic construction of quadratic integrals
and their involutivity. It secondly requires explaining the subtle way in which
suitably constructed canonical transformations reduce the Hamilton-Jacobi
problem of the (a priori time-dependent) driven part of the system into that of
an equivalent autonomous system of St\"ackel type
Lifted tensors and Hamilton-Jacobi separability
Starting from a bundle E over R, the dual of the first jet bundle, which is a
co-dimension 1 sub-bundle of the cotangent bundle of E, is the appropriate
manifold for the geometric description of time-dependent Hamiltonian systems.
Based on previous work, we recall properties of the complete lifts of a type
(1,1) tensor R on E to both of these manifolds. We discuss how an interplay
between these lifted tensors leads to the identification of related
distributions on both manifolds. The integrability of these distributions, a
coordinate free condition, is shown to produce exactly Forbat's conditions for
separability of the time-dependent Hamilton-Jacobi equation in appropriate
coordinates
The inverse problem for Lagrangian systems with certain non-conservative forces
We discuss two generalizations of the inverse problem of the calculus of
variations, one in which a given mechanical system can be brought into the form
of Lagrangian equations with non-conservative forces of a generalized Rayleigh
dissipation type, the other leading to Lagrangian equations with so-called
gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free
conditions for the existence of a suitable non-singular multiplier matrix,
which will lead to an equivalent representation of a given system of
second-order equations as one of these Lagrangian systems with non-conservative
forces.Comment: 28 page
Note on generalised connections and affine bundles
We develop an alternative view on the concept of connections over a vector
bundle map, which consists of a horizontal lift procedure to a prolonged
bundle. We further focus on prolongations to an affine bundle and introduce the
concept of affineness of a generalised connection.Comment: 17 page
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
Time-dependent kinetic energy metrics for Lagrangians of electromagnetic type
We extend the results obtained in a previous paper about a class of
Lagrangian systems which admit alternative kinetic energy metrics to
second-order mechanical systems with explicit time-dependence. The main results
are that a time-dependent alternative metric will have constant eigenvalues,
and will give rise to a time-dependent coordinate transformation which
partially decouples the system
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 (Q,g) and the way they give rise to a Poisson-Nijenhuis structure on the tangent bundle TQ. We then address the question of generalizing this concept to a Finsler space, where the metric tensor field comes from a regular Lagrangian function E, homogeneous of degree two in the fibre coordinates on TQ. It is shown that when a symmetric type (1,1) tensor field K along the tangent bundle projection τ: TQ→ Q 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.
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Lie algebroid structures on a class of affine bundles
We introduce the notion of a Lie algebroid structure on an affine bundle
whose base manifold is fibred over the real numbers. It is argued that this is
the framework which one needs for coming to a time-dependent generalization of
the theory of Lagrangian systems on Lie algebroids. An extensive discussion is
given of a way one can think of forms acting on sections of the affine bundle.
It is further shown that the affine Lie algebroid structure gives rise to a
coboundary operator on such forms. The concept of admissible curves and
dynamical systems whose integral curves are admissible, brings an associated
affine bundle into the picture, on which one can define in a natural way a
prolongation of the original affine Lie algebroid structure.Comment: 28 page
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