81,108 research outputs found
Classical Supersymmetric Mechanics
We analyse a supersymmetric mechanical model derived from (1+1)-dimensional
field theory with Yukawa interaction, assuming that all physical variables take
their values in a Grassmann algebra B. Utilizing the symmetries of the model we
demonstrate how for a certain class of potentials the equations of motion can
be solved completely for any B. In a second approach we suppose that the
Grassmann algebra is finitely generated, decompose the dynamical variables into
real components and devise a layer-by-layer strategy to solve the equations of
motion for arbitrary potential. We examine the possible types of motion for
both bosonic and fermionic quantities and show how symmetries relate the former
to the latter in a geometrical way. In particular, we investigate oscillatory
motion, applying results of Floquet theory, in order to elucidate the role that
energy variations of the lower order quantities play in determining the
quantities of higher order in B.Comment: 29 pages, 2 figures, submitted to Annals of Physic
Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem
The quasi-geostrophic two-layer model is of superior interest in dynamic
meteorology since it is one of the easiest ways to study baroclinic processes
in geophysical fluid dynamics. The complete set of point symmetries of the
two-layer equations is determined. An optimal set of one- and two-dimensional
inequivalent subalgebras of the maximal Lie invariance algebra is constructed.
On the basis of these subalgebras we exhaustively carry out group-invariant
reduction and compute various classes of exact solutions. Where possible,
reference to the physical meaning of the exact solutions is given. In
particular, the well-known baroclinic Rossby wave solutions in the two-layer
model are rediscovered.Comment: Extended version, 24 pages, 1 figur
Top-stable degenerations of finite dimensional representations I
Given a finite dimensional representation of a finite dimensional
algebra, two hierarchies of degenerations of are analyzed in the context of
their natural orders: the poset of those degenerations of which share the
top with - here denotes the radical of the algebra - and the
sub-poset of those which share the full radical layering
with . In particular, the article
addresses existence of proper top-stable or layer-stable degenerations - more
generally, it addresses the sizes of the corresponding posets including bounds
on the lengths of saturated chains - as well as structure and classification
Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schr\"odinger Equation in Two Dimensions
An exactly solvable position-dependent mass Schr\"odinger equation in two
dimensions, depicting a particle moving in a semi-infinite layer, is
re-examined in the light of recent theories describing superintegrable
two-dimensional systems with integrals of motion that are quadratic functions
of the momenta. To get the energy spectrum a quadratic algebra approach is used
together with a realization in terms of deformed parafermionic oscillator
operators. In this process, the importance of supplementing algebraic
considerations with a proper treatment of boundary conditions for selecting
physical wavefunctions is stressed. Some new results for matrix elements are
derived. This example emphasizes the interest of a quadratic algebra approach
to position-dependent mass Schr\"odinger equations.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
- …