47 research outputs found
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
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
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
Casimir Forces between Spherical Particles in a Critical Fluid and Conformal Invariance
Mesoscopic particles immersed in a critical fluid experience long-range
Casimir forces due to critical fluctuations. Using field theoretical methods,
we investigate the Casimir interaction between two spherical particles and
between a single particle and a planar boundary of the fluid. We exploit the
conformal symmetry at the critical point to map both cases onto a highly
symmetric geometry where the fluid is bounded by two concentric spheres with
radii R_- and R_+. In this geometry the singular part of the free energy F only
depends upon the ratio R_-/R_+, and the stress tensor, which we use to
calculate F, has a particularly simple form. Different boundary conditions
(surface universality classes) are considered, which either break or preserve
the order-parameter symmetry. We also consider profiles of thermodynamic
densities in the presence of two spheres. Explicit results are presented for an
ordinary critical point to leading order in epsilon=4-d and, in the case of
preserved symmetry, for the Gaussian model in arbitrary spatial dimension d.
Fundamental short-distance properties, such as profile behavior near a surface
or the behavior if a sphere has a `small' radius, are discussed and verified.
The relevance for colloidal solutions is pointed out.Comment: 37 pages, 2 postscript figures, REVTEX 3.0, published in Phys. Rev. B
51, 13717 (1995
Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries
We state the intrinsic form of the Hamiltonian equations of first-order
Classical Field theories in three equivalent geometrical ways: using
multivector fields, jet fields and connections. Thus, these equations are given
in a form similar to that in which the Hamiltonian equations of mechanics are
usually given. Then, using multivector fields, we study several aspects of
these equations, such as the existence and non-uniqueness of solutions, and the
integrability problem. In particular, these problems are analyzed for the case
of Hamiltonian systems defined in a submanifold of the multimomentum bundle.
Furthermore, the existence of first integrals of these Hamiltonian equations is
considered, and the relation between {\sl Cartan-Noether symmetries} and {\sl
general symmetries} of the system is discussed. Noether's theorem is also
stated in this context, both the ``classical'' version and its generalization
to include higher-order Cartan-Noether symmetries. Finally, the equivalence
between the Lagrangian and Hamiltonian formalisms is also discussed.Comment: Some minor mistakes are corrected. Bibliography is updated. To be
published in J. Phys. A: Mathematical and Genera
A non-linear Oscillator with quasi-Harmonic behaviour: two- and -dimensional Oscillators
A nonlinear two-dimensional system is studied by making use of both the
Lagrangian and the Hamiltonian formalisms. The present model is obtained as a
two-dimensional version of a one-dimensional oscillator previously studied at
the classical and also at the quantum level. First, it is proved that it is a
super-integrable system, and then the nonlinear equations are solved and the
solutions are explicitly obtained. All the bounded motions are quasiperiodic
oscillations and the unbounded (scattering) motions are represented by
hyperbolic functions. In the second part the system is generalized to the case
of degrees of freedom. Finally, the relation of this nonlinear system with
the harmonic oscillator on spaces of constant curvature, two-dimensional sphere
and hyperbolic plane , is discussed.Comment: 30 pages, 4 figures, submitted to Nonlinearit
Canonical quantization of so-called non-Lagrangian systems
We present an approach to the canonical quantization of systems with
equations of motion that are historically called non-Lagrangian equations. Our
viewpoint of this problem is the following: despite the fact that a set of
differential equations cannot be directly identified with a set of
Euler-Lagrange equations, 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. It turns out that in
the general case the hamiltonization and canonical quantization of such an
action are non-trivial problems, since the theory involves time-dependent
constraints. We adopt the general approach of hamiltonization and canonical
quantization for such theories (Gitman, Tyutin, 1990) to the case under
consideration. 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 proposed scheme is
applied to the quantization of a general quadratic theory. In addition, we
consider the quantization of a damped oscillator and of a radiating point-like
charge.Comment: 13 page
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