Approximation in a damped Hamiltonian system by successive relative equilibria

In this paper a (N+1)-dimensional Hamiltonian system with a cyclic coordinate and uniform friction added is considered. With the aid of the constant of motion of the system without friction, induced by the cyclic coordinate, we define a collection of special solutions of the nondissipative system, called the manifold of relataive equilibria (MRE), depending on 2 parameters. For some classes of Hamiltonians it is possible to approximate the solutions of the dissipative system that started in the neighbourhood of the MRE with the points of the MRE (hence with 2 parameters). We’ll show that the spherical pendulum is such a Hamiltonian system. Central in the analysis is that the dissipative autonomous problem is transformed to a Hamiltonian non-autonomous one. For such a system we prove under certain assumptions a generalisation of the well-known fact that the minimum of an autonomous Hamiltonian system is stable.