Multiseparability and Superintegrability for Classical and Q(7)

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

and

Then one can verify that Lk; H]= 0; Lk; Lj]= 0: How does one nd all orthogonal separable coordinate systems q for a given space Rn for zero potential, V 0? This is a di cult problem in di erential geometry. The answer is known for some constant curvature spaces. In real Euclidean 2-space there are four separable systems: cartesian, polar, parabolic and elliptic. For complex Euclidean 2-space, including real Euclidean space and real Minkowski space, there are six 4, 16, 18]: Cartesian, polar, parabolic, elliptic, hyperbolic and semi-hyperbolic. We describe these coordinate systems and their corresponding free particle constants of the motion L. (We adopt the basis px; py; M= xpy? ypx for the Lie algebra e(2; C ) and de ne p= px ipy,^.) There is one orbit of constants of the motion, with representative Mp, that is not associated with variable separation 21]. The separable systems are:+

(q)=

n k=1

( )

i

(qi ):

Cartesian coordinates

Polar Coordinates

x; y;

L= px2

(14)2

Parabolic Coordinates. 1 xP= 2 (? ); yP=; L= Mpy Elliptic Coordinates (in algebraic form) xE= c (u? 1)(v? 1); yE=?c uv;2 2 2 2 2 2

x= r cos; y= r sin;

L=M

(15) (16) (17)

Hyperbolic Coordinates2 2

L= M+ c px2 2 2 2 2

s s xH= r+ r rs+ s; yH= i r? r rs+ s; 2 2 L= M+p2 2 2 2 2 2+

(18)