The invariant manifolds of a finite straight segment

We give a method to compute explicitly the asymptotic expressions of some invariant manifolds in the vicinity of the collinear equilibrium points corresponding to the Hamiltonian system defined by the motion of a particle orbiting around a finite straight segment. For this we use normal form methods so that we make two different Lie transformations of the original Hamiltonian: (i) either we calculate the Hamilton function corresponding to the centre manifold of one of the collinear equilibrium points or (ii) either we determine the Hamiltonian related to the stable– unstable direction. By means of (i) we are able to parametrise the centre, the stable and the unstable manifolds of the original system using the direct changes of coordinates of the two transformations. Using (ii) we compute some 2D–tori and quasiperiodic orbits.