  This work originates from a first year undergraduate research project on
hidden symmetries of the dynamics for classical Hamiltonian systems, under the
program 'Jovens talentos para a Ciencia' of Brazilian funding agency Capes. For
pedagogical reasons the main subject chosen was Kepler's problem of motion
under a central potential, since it is a completely solved system. It is well
known that for this problem the group of dynamical symmetries is strictly
larger than the isometry group O(3), the extra symmetries corresponding to
hidden symmetries of the dynamics. By taking the point of view of examining the
group action of the dynamical symmetries on the allowed trajectories, it is
possible to teach in the same project basic elements of as many important
subjects in physics as: Hamiltonian formalism, hidden symmetries, integrable
systems, group theory, and the use of manifolds.
