Separation of variables for the Hamilton-Jacobi equation on complex projective spaces

Abstract

The additive separation of variables in the Hamilton-Jacobi equation and the multiplicative separation of variables in the Laplace-Beltrami equation are studied for the complex projective space C Pⁿ considered as a Riemannian Einstein space with the standard Fubini-Study metric. The isometry group of C Pⁿ is SU(ⁿ + 1) and its Cartan subgroup is used to generate n ignorable variables (variables not figuring in the metric tensor). A one-to-one correspondence is established between separable coordinate systems on S and separable systems with n ignorable variables on C P. The separable coordinates in C Pⁿ are characterized by 2n integrals of motion in involution: n of them are elements of the Cartan subalgebra of SU(n + 1) and the remaining n are linear combinations of the Casimir operators of n(n + 1)/2 different su(2) subalgebras of su(n + 1). Each system of 2n integrals of motion in involution, and hence each separable system of coordinates on CPⁿ, thus provides a completely integrable Hamiltonian system. For n= 2 it is shown that only two separable systems on CP² exist, both nonorthogonal with two ignorable variables, coming from spherical and elliptic coordinates on S², respectively.