Jacobi method and the eikonal equation (Q1128383)

The first part of the paper contains a discussion of the analogy between the Hamilton-Jacobi equation modelling the rotation of a rigid body around a fixed point and some variants of the eikonal equation in optics. Next, a survey of the best known integrability cases (Euler's, Lagrange's, Kowalevskaya's, Delone's, Hess') of the equation of a rotating rigid body is presented. The last part of the paper concerns the so called ``group analysis'' of the canonical differential system associated to a smooth Hamiltonian where the following Proposition is proved: ``The Hamiltonian symmetry leading to the construction of an integral of the canonical equations is in general not a dynamical symmetry''.