Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

Abstract

We describe Jacobi's method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrödinger operator with potential on a Riemannian space is second-order superintegrable if there are 2n − 1 (classically) functionally independent second-order symmetry operators. (The 2n − 1 is the maximum possible number of such symmetries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general structure of superintegrable systems in all real or complex two-dimensional spaces and three-dimensional conformally flat spaces and a complete list of such spaces and potentials in two dimensions.

(Non)-Integrability of Geodesics in D-brane Backgrounds

“…In particular, reference [9] demonstrated that deformation of AdS 5 × S 5 to asymptotically-flat geometry by adding one to the harmonic function destroys integrability of sigma model, although the Hamilton-Jacobi equation for geodesics remains separable. The extension from AdS 5 × S 5 to flat geometry is only possible if one choses flat metric on the worldvolume of D3 branes 31 , and in section 3 we analyzed several classes of geometries produced by flat Dp branes. From the point of view of AdS/CFT correspondence, it is equally interesting to look at field theories on R × S 3 , which are dual to geometries produced by spherical D3 branes.…”

(Non)-integrability of geodesics in D-brane backgrounds

“…then equation (B.42) ensures the symmetry of H. 45 Modifying the arguments that led to (B.22), we conclude that…”

Hidden symmetries in string theory

“…As our final example of elliptic coordinates, we consider AdS 3 ×S 3 in global parameterization, which can be obtained by taking the near horizon limit (H → Q/f ) in (2.169) [63]: 37 We redefined angle θ in (2.199). Alternatively, one can keep (2.199) and shift θ in (2.74).…”

“…37) There are p terms in this equation. Using the defining relation(3.15) for the CKYT, ∇ b Y acd... = −∇ a Y bcd... + 2g ab Z cd... − [g ca Z bd... + g cb Z ad... ] + .…”