454 research outputs found
Integrable and superintegrable systems with spin
A system of two particles with spin s=0 and s=1/2 respectively, moving in a
plane is considered. It is shown that such a system with a nontrivial
spin-orbit interaction can allow an 8 dimensional Lie algebra of first-order
integrals of motion. The Pauli equation is solved in this superintegrable case
and reduced to a system of ordinary differential equations when only one
first-order integral exists.Comment: 12 page
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
Construction of classical superintegrable systems with higher order integrals of motion from ladder operators
We construct integrals of motion for multidimensional classical systems from
ladder operators of one-dimensional systems. This method can be used to obtain
new systems with higher order integrals. We show how these integrals generate a
polynomial Poisson algebra. We consider a one-dimensional system with third
order ladders operators and found a family of superintegrable systems with
higher order integrals of motion. We obtain also the polynomial algebra
generated by these integrals. We calculate numerically the trajectories and
show that all bounded trajectories are closed.Comment: 10 pages, 4 figures, to appear in j.math.phys
Superintegrability in a two-dimensional space of nonconstant curvature
A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton–Jacobi and Schrödinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined
Quantum superintegrability and exact solvability in N dimensions
A family of maximally superintegrable systems containing the Coulomb atom as
a special case is constructed in N-dimensional Euclidean space. Two different
sets of N commuting second order operators are found, overlapping in the
Hamiltonian alone. The system is separable in several coordinate systems and is
shown to be exactly solvable. It is solved in terms of classical orthogonal
polynomials. The Hamiltonian and N further operators are shown to lie in the
enveloping algebra of a hidden affine Lie algebra
Superintegrable Systems in Darboux spaces
Almost all research on superintegrable potentials concerns spaces of constant
curvature. In this paper we find by exhaustive calculation, all superintegrable
potentials in the four Darboux spaces of revolution that have at least two
integrals of motion quadratic in the momenta, in addition to the Hamiltonian.
These are two-dimensional spaces of nonconstant curvature. It turns out that
all of these potentials are equivalent to superintegrable potentials in complex
Euclidean 2-space or on the complex 2-sphere, via "coupling constant
metamorphosis" (or equivalently, via Staeckel multiplier transformations). We
present tables of the results
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
Maximal Abelian Subgroups of the Isometry and Conformal Groups of Euclidean and Minkowski Spaces
The maximal Abelian subalgebras of the Euclidean e(p,0) and pseudoeuclidean
e(p,1)Lie algebras are classified into conjugacy classes under the action of
the corresponding Lie groups E(p,0) and E(p,1), and also under the conformal
groups O(p+1,1) and O(p+1,2), respectively. The results are presented in terms
of decomposition theorems. For e(p,0) orthogonally indecomposable MASAs exist
only for p=1 and p=2. For e(p,1), on the other hand, orthogonally
indecomposable MASAs exist for all values of p. The results are used to
construct new coordinate systems in which wave equations and Hamilton-Jacobi
equations allow the separation of variables.Comment: 31 pages, Latex (+ latexsym
- …