85 research outputs found
Frobenius theorem and invariants for Hamiltonian systems
We apply Frobenius integrability theorem in the search of invariants for
one-dimensional Hamiltonian systems with a time-dependent potential. We obtain
several classes of potential functions for which Frobenius theorem assures the
existence of a two-dimensional foliation to which the motion is constrained. In
particular, we derive a new infinite class of potentials for which the motion
is assurately restricted to a two-dimensional foliation. In some cases,
Frobenius theorem allows the explicit construction of an associated invariant.
It is proven the inverse result that, if an invariant is known, then it always
can be furnished by Frobenius theorem
Analytic Solution of Emden-Fowler Equation and Critical Adsorption in Spherical Geometry
In the framework of mean-field theory the equation for the order-parameter
profile in a spherically-symmetric geometry at the bulk critical point reduces
to an Emden-Fowler problem. We obtain analytic solutions for the surface
universality class of extraordinary transitions in for a spherical shell,
which may serve as a starting point for a pertubative calculation. It is
demonstrated that the solution correctly reproduces the Fisher-de Gennes effect
in the limit of the parallel-plate geometry.Comment: (to be published in Z. Phys. B), 7 pages, 1 figure, uuencoded
postscript file, 8-9
Noether symmetries for two-dimensional charged particle motion
We find the Noether point symmetries for non-relativistic two-dimensional
charged particle motion. These symmetries are composed of a quasi-invariance
transformation, a time-dependent rotation and a time-dependent spatial
translation. The associated electromagnetic field satisfy a system of
first-order linear partial differential equations. This system is solved
exactly, yielding three classes of electromagnetic fields compatible with
Noether point symmetries. The corresponding Noether invariants are derived and
interpreted
Autonomous three dimensional Newtonian systems which admit Lie and Noether point symmetries
We determine the autonomous three dimensional Newtonian systems which admit
Lie point symmetries and the three dimensional autonomous Newtonian Hamiltonian
systems, which admit Noether point symmetries. We apply the results in order to
determine the two dimensional Hamiltonian dynamical systems which move in a
space of constant non-vanishing curvature and are integrable via Noether point
symmetries. The derivation of the results is geometric and can be extended
naturally to higher dimensions.Comment: Accepted for publication in Journal of Physics A: Math. and Theor.,13
page
On the action principle for a system of differential equations
We consider the problem of constructing an action functional for physical
systems whose classical equations of motion cannot be directly identified with
Euler-Lagrange equations for an action principle. Two ways of action principle
construction are presented. From simple consideration, we derive necessary and
sufficient conditions for the existence of a multiplier matrix which can endow
a prescribed set of second-order differential equations with the structure of
Euler-Lagrange equations. An explicit form of the action is constructed in case
if such a multiplier exists. If a given set of differential equations cannot be
derived from an action principle, one can reformulate such a set in an
equivalent first-order form which can always be treated as the Euler-Lagrange
equations of a certain action. We construct such an action explicitly. There
exists an ambiguity (not reduced to a total time derivative) in associating a
Lagrange function with a given set of equations. We present a complete
description of this ambiguity. The general procedure is illustrated by several
examples.Comment: 10 page
A class of Poisson-Nijenhuis structures on a tangent bundle
Equipping the tangent bundle TQ of a manifold with a symplectic form coming
from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis
structure from a given type (1,1) tensor field J on Q. It is argued that the
complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ,
but plays a crucial role in the construction of a different tensor R, which
appears to be the pullback under the Legendre transform of the lift of J to
co-tangent manifold of Q. We show how this tangent bundle view brings new
insights and is capable also of producing all important results which are known
from previous studies on the cotangent bundle, in the case that Q is equipped
with a Riemannian metric. The present approach further paves the way for future
generalizations.Comment: 22 page
Higher-order Abel equations: Lagrangian formalism, first integrals and Darboux polynomials
A geometric approach is used to study a family of higher-order nonlinear Abel
equations. The inverse problem of the Lagrangian dynamics is studied in the
particular case of the second-order Abel equation and the existence of two
alternative Lagrangian formulations is proved, both Lagrangians being of a
non-natural class (neither potential nor kinetic term). These higher-order Abel
equations are studied by means of their Darboux polynomials and Jacobi
multipliers. In all the cases a family of constants of the motion is explicitly
obtained. The general n-dimensional case is also studied
Lie symmetries for two-dimensional charged particle motion
We find the Lie point symmetries for non-relativistic two-dimensional charged
particle motion. These symmetries comprise a quasi-invariance transformation, a
time-dependent rotation, a time-dependent spatial translation and a dilation.
The associated electromagnetic fields satisfy a system of first-order linear
partial differential equations. This system is solved exactly, yielding four
classes of electromagnetic fields compatible with Lie point symmetries
The Inverse Variational Problem for Autoparallels
We study the problem of the existence of a local quantum scalar field theory
in a general affine metric space that in the semiclassical approximation would
lead to the autoparallel motion of wave packets, thus providing a deviation of
the spinless particle trajectory from the geodesics in the presence of torsion.
The problem is shown to be equivalent to the inverse problem of the calculus of
variations for the autoparallel motion with additional conditions that the
action (if it exists) has to be invariant under time reparametrizations and
general coordinate transformations, while depending analytically on the torsion
tensor. The problem is proved to have no solution for a generic torsion in
four-dimensional spacetime. A solution exists only if the contracted torsion
tensor is a gradient of a scalar field. The corresponding field theory
describes coupling of matter to the dilaton field.Comment: 13 pages, plain Latex, no figure
Classical field theory on Lie algebroids: Variational aspects
The variational formalism for classical field theories is extended to the
setting of Lie algebroids. Given a Lagrangian function we study the problem of
finding critical points of the action functional when we restrict the fields to
be morphisms of Lie algebroids. In addition to the standard case, our formalism
includes as particular examples the case of systems with symmetry (covariant
Euler-Poincare and Lagrange Poincare cases), Sigma models or Chern-Simons
theories.Comment: Talk deliverd at the 9th International Conference on Differential
Geometry and its Applications, Prague, September 2004. References adde
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