Lagrangian actions for elliptical solutions of 2-body and 3-body problems with fixed energies

Based on the works of Gordon (1977) and Zhang and Zhou (2001) on the variational minimizing properties for Keplerian orbits and Lagrangian solutions of Newtonian 2-body and 3-body problems, we use the constrained variational principle of Ambrosetti and Coti Zelati (1990) to compute the Lagrangian actions on Keplerian and Lagrangian elliptical solutions with fixed energies. We also find an interesting relation between the period and the energy for Lagrangian elliptical solutions with Newtonian potentials.