Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory

Abstract

This paper is part of a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in real or complex conformally flat spaces. Here we consider classical superintegrable systems with nondegenerate potentials in three dimensions. We show that there exists a standard structure for such systems, based on the algebra of 3 ϫ 3 symmetric matrices, and that the quadratic algebra always closes at order 6. We show that the spaces of truly second-, third-, fourth-, and sixth-order constants of the motion are of dimension 6, 4, 21, and 56, respectively, and we construct explicit bases for the fourth-and sixth order constants in terms of products of the second order constants.

Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

“…[5][6][7] Like much of the later work on superintegrable systems, it was restricted to the case of second-order integrals of motion. [8][9][10][11][12][13] This case turned out to have an intimate connection with the separation of variables in the HamiltonJacobi and Schrödinger equations.…”

Second Order Superintegrable Systems in Three Dimensions

“…However, in the 3D case there are only 5 functionally independent symmetries, so we can't guarantee that the symmetry equations admit a 6-parameter family of solutions. Fortunately, by careful study of the integrability conditions of these equations and use of the requirement that the potential is nondegenerate, we can prove the 5 =⇒ 6 theorem [20].…”

“…This means that we require that the five quadratic forms L h , H 0 are functionally independent.) In [20] it is shown that the matrix of the 15 Bertrand-Darboux equations for the potential has rank at least 5, hence we can solve for the second derivatives of the potential in the form…”

“…Two dimensional second order superintegrable systems have been studied and classified by the author and his collaborators in a recent series of papers [18,19,20,21]. Here we concentrate on three dimensional (3D) systems where new complications arise.…”

“…The key difference with our first example is, as we shall show later, that the 3-parameter Kepler-Coulomb potential is degenerate and it cannot be extended to a 4-parameter potential. In [20,21] there are examples of superintegrable systems on the 3-sphere that admit a quadratic algebra structure. A more general set of examples arises from a space with metric…”