Dynamical groups and spherical potentials in Classical Mechanics

Abstract

The one particle problem in a spherical potential is examined in Classical Mechanics from a group theorical point of view. The constants of motion are classified according to their behaviour under the rotation group $0(3), i.e. according to the irreducible representations D 0 of 80(3) (section 1).The Lie algebras of SO(4) and 80(3) are explicitly built in terms of Poisson brackets for an arbitrary potential, from global considerations. The Kepler and the 3 dimensional oscillator problems are shown to play particular roles with respect to these groups (sections 2 and 3).In the last section, the Kepler problem is analyzed with the aid of the $0(4) group instead of the Lie algebra. It is proved that the transformations generated by the angular momentum and the Runge-Lenz vector form indeed a group of canonical transformations isomorphic to $0(4). Consequences with respect to the quantization problem are examined.

Higgs Algebraic Symmetry of Screened System in a Spherical Geometry

“…The algebraic relations of their conserved quantities reveal the dynamical symmetries of the two systems are described by the symmetry groups SO(3) and SU (2) respectively. This algebraic structure is called Higgs Algebra, and has received attention from a variety of literatures [14,15,16,17,18,19,20]. The Bertrand's theorem has been extended in different directions [21,22,23,24].…”

On Peres approach to Fradkin-Bacry-Ruegg-Souriau’s perihelion vector

“…Bacry, Ruegg, Souriau [12] and Fradkin [13] made an important additional step when they showed that all three dimensional dynamical problems involving central potentials possess the extended symmetry algebra O 4 and SU 3 , a result subsequently generalized by Mukunda [14] for any Hamiltonian system with n degrees of freedom. As noted by Bacry, Ruegg and Souriau in their seminal paper [12], the general ability to construct such a local Lie algebra of dynamical symmetry is not surprising since all the 2n-dimensional symplectic manifolds are locally isomorphic [15].…”

“…As noted by Bacry, Ruegg and Souriau in their seminal paper [12], the general ability to construct such a local Lie algebra of dynamical symmetry is not surprising since all the 2n-dimensional symplectic manifolds are locally isomorphic [15]. However the subsequent step to ensure a global dynamical symmetry is to verify that the finite canonical transformations generated by this algebra form a group, a feature of the Kepler problem.…”

“…Bacry, Ruegg, Souriau [12] and Fradkin [13] gave an explicit construction scheme for a vector with constancy properties associated with every central potential, Fradkin's derivation being very direct and explicit. Bacry, Ruegg and Souriau already noted that such a vector is exceptionally single-valued, as in the Kepler case.…”

Illustrating dynamical symmetries in classical mechanics: The Laplace–Runge–Lenz vector revisited

“…There are hundreds of papers [1][2][3][4][5][6][7][8][9][10] on the power, applications, and extensions of the Laplace-Runge-Lenz vector. The paper by Mukunda 11 comes closest to what we discuss below, but our aim is primarily pedagogical.…”