Geometric Hamilton–Jacobi theory

Abstract. The Hamilton–Jacobi problem is revisited having in mind the con-sequences arising from a possible bi-Hamiltonian structure. To avoid the bias of the existence of a natural symplectic structure on the cotangent bundle, the problem is formulated on the tangent bundle for Lagrangian systems. The problem is first developed for systems described by regular Lagrangians and extended to systems described by singular Lagrangians with no secondary con-straints. We also consider the examples of the free relativistic particle, free motion on a Lie group, the rigid body and the electron-monopole system. 1