A generalization of Gordon's Theorem to arbitrary winding number

Abstract

We study minimizers of the Keplerian action functional with homogeneous potential of type $1/|u{|}^{\alpha }$ on periodic loops with fixed winding number $k\in \mathbb{Z}\setminus \{0\}$. This extends the classical theorems of Gordon and Venturelli, closing the previously open case $\alpha \in (1,2)$, $|k|\ge 2$. We show that the $k$-circular solution loses local minimality below the threshold $\alpha =2-1/|k{|}^2$, and that minimizers are collision-free above $\alpha =2-1/|k|$. Combining these results with the existence of precessing periodic orbits in between, we obtain a full classification of the minimizer in terms of $\alpha$ and $k$, which we also verify numerically for $k=2$.