A generalization of Gordon's Theorem to arbitrary winding number

Technical Summary: A Generalization of Gordon's Theorem to Arbitrary Winding Number

Problem Statement
The paper addresses the minimization of the Keplerian action functional with a homogeneous potential of type $1/|u|^\alpha$ on periodic loops with a fixed non-zero winding number $k \in \mathbb{Z} \setminus \{0\}$. The action functional is defined as:
$$A_\alpha(u) = \int_0^T \left( \frac{1}{2} |\dot{u}|^2 + \frac{1}{|u|^\alpha} \right) dt$$
where $u: [0, T] \to \mathbb{R}^2$ is a periodic loop in the Sobolev space $H^1$.

The problem builds upon classical results by Gordon and Venturelli regarding the "weak force" case ($0 < \alpha < 2$). Gordon established that for $\alpha=1$, minimizers are collision-ejection solutions for $|k| \ge 2$, while Venturelli generalized this to $\alpha \in (0, 1)$ for all $k$. For $\alpha \in (1, 2)$ and $|k|=1$, the minimizer is known to be the circular solution. However, the case for $\alpha \in (1, 2)$ with $|k| \ge 2$ remained open. The difficulty in this regime arises because the circular solution can lose local minimality (develop conjugate points), and new families of periodic solutions—specifically precessing orbits that oscillate radially while winding $k$ times—enter the competition for the global minimum.

Methodology
The author employs a combination of analytical techniques and numerical verification to classify the minimizers across the parameter space $(\alpha, k)$. The approach relies on three primary analytical ingredients:

1. Jacobi Field Analysis: The paper explicitly computes Jacobi fields around the $k$-circular solution to determine the existence of conjugate points. By analyzing the determinant of the fundamental solution matrix, the author derives the precise threshold where the circular solution ceases to be a local minimizer.
2. Blow-up Technique: To address the presence of collisions, the author adapts the blow-up technique (previously used by Venturelli) to establish a lower bound on $\alpha$ required for minimizers to be collision-free. This involves analyzing the behavior of minimizing sequences near the singularity and estimating the precession angle swept during a collision-ejection event.
3. Continuity Argument for Precessing Orbits: The existence of periodic precessing solutions (specifically $q$-precessing motions with $q$ radial oscillations per period) is proven within a specific intermediate range of $\alpha$. This relies on the continuity of the precession angle $\Phi(h, c)$ with respect to energy $h$ and angular momentum $c$, showing that orbits with the required winding number $k$ exist when the precession angle matches $\pi|k|$.
4. Numerical Verification: For the specific case of $k=2$, the author performs numerical computations using truncated Fourier series, gradient descent, and shooting methods to compute the action values of the three competing solution families (collision-ejection, precessing, and circular) and verify the theoretical transition points.

Key Results and Classification
The paper establishes a complete classification of the minimizer of $A_\alpha$ on the space of loops with winding number $k$, dependent on the value of $\alpha$ relative to two critical thresholds: $\alpha_1 = 2 - 1/|k|$ and $\alpha_2 = 2 - 1/|k|^2$.

• Regime I: $\alpha \in (1, 2 - 1/|k|]$
  In this range, the minimizer is the collision-ejection solution. The circular solution is not a minimizer due to the presence of conjugate points (for $|k| \ge 2$), and precessing solutions either do not exist or possess higher action than the collision-ejection solution.

• Regime II: $\alpha \in (2 - 1/|k|, 2 - 1/|k|^2)$
  In this intermediate range, the minimizer is a periodic precessing solution (specifically a $q$-precessing motion). The minimizer is collision-free. For $|k| = 2, 3, 4$, only $1$-precessing motions exist in this range. For $|k| \ge 5$, higher-order precessing motions ($q \ge 2$) may also exist, though the paper does not definitively identify which specific $q$-family minimizes the action in all sub-cases.

• Regime III: $\alpha \in [2 - 1/|k|^2, 2)$
  In this range, the minimizer is the $k$-circular solution. Here, the circular orbit is free of conjugate points, and no precessing solutions exist that satisfy the winding number constraint within the period.

Significance and Claims
The paper claims to close the previously open case of $\alpha \in (1, 2)$ for arbitrary winding numbers $|k| \ge 2$, extending the classical theorems of Gordon and Venturelli. The significance lies in the full classification of minimizers, which reveals a rich structure where the nature of the minimizer transitions from singular (collision-ejection) to intermediate (precessing) to regular (circular) as the potential strength $\alpha$ increases.

The author notes that while the classification is complete regarding the types of solutions (collision-ejection, precessing, circular), a finer classification remains open for $|k| \ge 5$. In this regime, multiple families of $q$-precessing solutions coexist in the intermediate range, and the paper does not resolve which specific family acts as the global minimizer, nor does it rule out transitions between families as $\alpha$ varies. The numerical study for $k=2$ confirms the theoretical predictions, showing that the transition between minimizing regimes occurs exactly at the derived thresholds.