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One of the difficulties encountered when studying physical theories in discrete space-time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices. In a previous article we… (More)

- E. G. Kalnins, Jonathan M. KRESS, P. Winternitz
- 2008

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 twodimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical Email:… (More)

Invariants of the coadjoint representation of two classes of Lie algebras are calculated. The first class consists of the nilpotent Lie algebras T (M), isomorphic to the algebras of upper triangular M × M matrices. The Lie algebra T (M) is shown to have [M/2] functionally independent invariants. They can all be chosen to be polynomials and they are… (More)

We consider here the coexistence of firstand third-order integrals of motion in two dimensional classical and quantum mechanics. We find explicitly all potentials that admit such integrals, and all their integrals. Quantum superintegrable systems are found that have no classical analog, i.e. the potentials are proportional to h̄, so their classical limit is… (More)

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to several examples. The found symmetry groups are used to obtain particular solutions of differential-difference equations.

A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. They take solutions into solutions and can be used to perform symmetry reduction. The method generalizes one presented in a recent publication for the case of… (More)

The Lie symmetries of a large class of generalized Toda field theories are studied and used to perform symmetry reduction. Reductions lead to generalized Toda lattices on one hand, to periodic systems on the other. Boundary conditions are introduced to reduce theories on an infinite lattice to those on semi-infinite, or finite ones. E-mail:… (More)

- Miguel A. Rodríguez, Piergiulio Tempesta, P. Winternitz
- Physical review. E, Statistical, nonlinear, and…
- 2008

We introduce a 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… (More)

A review is given of some recently obtained results on analytic contractions of Lie algebras and Lie groups and their application to special function theory. The contractions considered are from O(3) to E(2) and from O(2, 1) to E(2), or E(1, 1). The analytic contractions provide relations between separable coordinate systems on various homogeneous… (More)

We show that the three body Calogero model with inverse square potentials can be interpreted as a maximally superintegrable and multiseparable system in Euclidean three-space. As such it is a special case of a family of systems involving one arbitrary function of one