Geodesic flows for the Neumann-Rosochatius systems

The Relationship between the Neumann system and the Jacobi system in arbitrary dimensions is elucidated from the point of view of constrained Hamiltonian systems. Dirac brackets for canonical variables of both systems are derived from the constrained Hamiltonians. The geodesic equations corresponding to the Rosochatius system are studied as an application of our method. As a consequence a new class of nonlinear integrable equations is derived along with their conserved quantities.