Abstract
In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. Here we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.
Highlights
We consider the classical Hamiltonian in RN given by: p2 mω2q2Hλ(q, p) = Tλ(q, p) + Uλ(q) = 2m(1 + λq2) + 2(1 + λq2), (1)where ω and λ are real positive parameters, q = (q1, . . . , qN ), p = (p1, . . . , pN ) ∈ RN are conjugate coordinates and momenta, q2 = N i=1 qi2 and = p2i .The kinetic energyTλ(q, p) can be interpreted as the Lagrangian generating the geodesic motion of a particle of mass m on a conformally flat space known as the Darboux space of type III (D-III)
We deal just with the classical system and restrict our considerations to the 3D case, with emphasis on the algebraic side of the problem. In this context we show that the so-called Spectrum Generating Algebra (SGA) technique [16, 17] provides us with all the necessary ingredients to achieve its solution, at least in the λ > 0 case, in full analogy with the classical Taub-NUT system [18]
In this way we have found r(t) and p(t) and the motion has been fully determined by means of the SGA procedure
Summary
We notice that in [6] the Hamiltonian Hλ has been proven to be maximally superintegrable by taking advantage of its super-separability (like the usual harmonic oscillator, the system turns out to be separable both in Cartesian and (hyper)spherical coordinates). This property has been further analysed in a number of subsequent papers (see, for instance, [11, 12, 13] and references therein), where its algebraic content was explained in terms of the Demkov– Fradkin tensor [14, 15] given by. The main results and some further perspectives are outlined in the concluding section
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