336 research outputs found
Classical Trajectories for two Ring-Shaped Potentials
This paper deals with the classical trajectories for two super-integrable
systems: a system known in quantum chemistry as the Hartmann system and a
system of potential use in quantum chemistry and nuclear physics. Both systems
correspond to ring-shaped potentials. They admit two maximally super-integrable
systems as limiting cases, viz, the isotropic harmonic oscillator system and
the Coulomb-Kepler system in three dimensions. The planarity of the
trajectories is studied in a systematic way. In general, the trajectories are
quasi-periodic rather than periodic. A constraint condition allows to pass from
quasi-periodic motions to periodic ones. When written in a quantum mechanical
context, this constraint condition leads to new accidental degeneracies for the
two systems studied.Comment: 28 pages, Tex fil
Third-order superintegrable systems separable in parabolic coordinates
In this paper, we investigate superintegrable systems which separate in
parabolic coordinates and admit a third-order integral of motion. We give the
corresponding determining equations and show that all such systems are
multi-separable and so admit two second-order integrals. The third-order
integral is their Lie or Poisson commutator. We discuss how this situation is
different from the Cartesian and polar cases where new potentials were
discovered which are not multi-separable and which are expressed in terms of
Painlev\'e transcendents or elliptic functions
Exact Solutions of a (2+1)-Dimensional Nonlinear Klein-Gordon Equation
The purpose of this paper is to present a class of particular solutions of a
C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry
reduction. Using the subgroups of similitude group reduced ordinary
differential equations of second order and their solutions by a singularity
analysis are classified. In particular, it has been shown that whenever they
have the Painlev\'e property, they can be transformed to standard forms by
Moebius transformations of dependent variable and arbitrary smooth
transformations of independent variable whose solutions, depending on the
values of parameters, are expressible in terms of either elementary functions
or Jacobi elliptic functions.Comment: 16 pages, no figures, revised versio
A class of anisotropic (Finsler-) space-time geometries
A particular Finsler-metric proposed in [1,2] and describing a geometry with
a preferred null direction is characterized here as belonging to a subclass
contained in a larger class of Finsler-metrics with one or more preferred
directions (null, space- or timelike). The metrics are classified according to
their group of isometries. These turn out to be isomorphic to subgroups of the
Poincar\'e (Lorentz-) group complemented by the generator of a dilatation. The
arising Finsler geometries may be used for the construction of relativistic
theories testing the isotropy of space. It is shown that the Finsler space with
the only preferred null direction is the anisotropic space closest to isotropic
Minkowski-space of the full class discussed.Comment: 12 pages, latex, no figure
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