145 research outputs found
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
Reduction of superintegrable systems: the anisotropic harmonic oscillator
We introduce a new 2N--parametric family of maximally superintegrable systems
in N dimensions, obtained as a reduction of an anisotropic harmonic oscillator
in a 2N--dimensional configuration space. These systems possess closed bounded
orbits and integrals of motion which are polynomial in the momenta. They
generalize known examples of superintegrable models in the Euclidean plane.Comment: 6 pages. Version accepted in Physical Review
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
Superintegrable Systems with a Third Order Integrals of Motion
Two-dimensional superintegrable systems with one third order and one lower
order integral of motion are reviewed. The fact that Hamiltonian systems with
higher order integrals of motion are not the same in classical and quantum
mechanics is stressed. New results on the use of classical and quantum third
order integrals are presented in Section 5 and 6.Comment: To appear in J. Phys A: Mathematical and Theoretical (SPE QTS5
Quantum Lie systems and integrability conditions
The theory of Lie systems has recently been applied to Quantum Mechanics and
additionally some integrability conditions for Lie systems of differential
equations have also recently been analysed from a geometric perspective. In
this paper we use both developments to obtain a geometric theory of
integrability in Quantum Mechanics and we use it to provide a series of
non-trivial integrable quantum mechanical models and to recover some known
results from our unifying point of view
Self-Adjointness of Generalized MIC-Kepler System
We have studied the self-adjointness of generalized MIC-Kepler Hamiltonian,
obtained from the formally self-adjoint generalized MIC-Kepler Hamiltonian. We
have shown that for \tilde l=0, the system admits a 1-parameter family of
self-adjoint extensions and for \tilde l \neq 0 but \tilde l <{1/2}, it has
also a 1-parameter family of self-adjoint extensions.Comment: 11 pages, Latex, no figur
Superintegrable Systems, Multi-Hamiltonian Structures and Nambu Mechanics in an Arbitrary Dimension
A general algebraic condition for the functional independence of 2n-1
constants of motion of an n-dimensional maximal superintegrable Hamiltonian
system has been proved for an arbitrary finite n. This makes it possible to
construct, in a well-defined generic way, a normalized Nambu bracket which
produces the correct Hamiltonian time evolution. Existence and explicit forms
of pairwise compatible multi-Hamiltonian structures for any maximal
superintegrable system have been established. The Calogero-Moser system, motion
of a charged particle in a uniform perpendicular magnetic field and
Smorodinsky-Winternitz potentials are considered as illustrative applications
and their symmetry algebras as well as their Nambu formulations and alternative
Poisson structures are presented.Comment: 20 pages, 1 table (submitted for publication
Point Symmetries of Generalized Toda Field Theories
A class of two-dimensional field theories with exponential interactions is
introduced. The interaction depends on two ``coupling'' matrices and is
sufficiently general to include all Toda field theories existing in the
literature. Lie point symmetries of these theories are found for an infinite,
semi-infinite and finite number of fields. Special attention is accorded to
conformal invariance and its breaking.Comment: 25 pages, no figures, Latex fil
Group-theoretical approach to a non-central extension of the Kepler-Coulomb problem
Bound and scattering states of a non-central extension of the
three-dimensional Kepler-Coulomb Hamiltonian are worked out analytically within
the framework of the potential groups of the problem, SO(7) for bound states
and SO(6,1) for scattering states. In the latter case, the S matrix is
calculated by the method of intertwining operators.Comment: 12 pages, to appear in J. Phys. A : Math. Theo
Solvable Lie algebras with triangular nilradicals
All finite-dimensional indecomposable solvable Lie algebras , having
the triangular algebra T(n) as their nilradical, are constructed. The number of
nonnilpotent elements in satisfies and the
dimension of the Lie algebra is
- …