Entire Minimal Parabolic Trajectories: The Planar Anisotropic Kepler Problem

We continue the variational approach to parabolic trajectories introduced in our previous paper (Barutello et al., Entire parabolic trajectories as minimal phase transitions. arXiv:1105.3358v1, 2011), which sees parabolic orbits as minimal phase transitions. We deepen and complete the analysis in the planar case for homogeneous singular potentials. We characterize all parabolic orbits connecting two minimal central configurations as free-time Morse minimizers (in a given homotopy class of paths). These may occur for at most one value of the homogeneity exponent. In addition, we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed-end problems, and also with the existence of action minimizing periodic trajectories with nontrivial homotopy type.