Separation of variables for the Hamilton-Jacobi equation on complex projective spaces
Boyer, C. P., Kalnins, E. G., & Winternitz, P. (1985). Separation of variables for the Hamilton-Jacobi equation on complex projective spaces. SIAM Journal on Mathematical Analysis, 16(1), 93–109. https://doi.org/10.1137/0516006
Permanent Research Commons link: https://hdl.handle.net/10289/12356
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.
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