The problem of the dynamics of colliding galaxies is considered. A first approximation for the motions of the centers of mass of colliding galaxies is developed. In this approximation, the galaxies are treated as spherically symmetric distributions of constituent point masses (the stars), and the effects of collisions are tentatively assumed to be negligible insofar as changes in the structures and internal energies of the galaxies are concerned. On the basis of these results, an approximate theory has been developed to provide estimates for the exchange of energy between that corresponding to the center-of-mass motions of colliding galaxies and that corresponding to their internal structures and motions. For given initial conditions, the results concerning the energy exchange provide not only a consistency test for the first approximation for the motions of the centers of mass but also approximate lower-limits for the actual energy exchange-regardless of the validity of this first approximation. From the results obtained, it appears that the effect of collisions on the internal energies of galaxies is considerably more important than may previously have been believed and has probably been important in altering the internal energies of non-negligible fraction of the galaxies in the observable universe.