[Paper Review] Choreography solutions of the $n$-body problem on $S^2$

Juan Manuel Sánchez-Cerritos, Shiqing Zhang|arXiv (Cornell University)|Jun 8, 2018

Spacecraft Dynamics and Control17 references3 citations

TL;DR

This paper proves the existence of choreography solutions for the three-body problem on the 2-sphere ($S^2$) using variational methods. By minimizing the Lagrangian action functional over a symmetric subset $H$ of periodic, non-colliding configurations, the authors establish a non-collision, periodic solution with an 8-shaped orbit, extending the classical planar eight-shape choreography to curved space with positive curvature.

ABSTRACT

We try to prove the existence of choreography solutions for the $n-$body problem on $S^2$. For the three-body problem, we show the existence of the 8-shape orbit on $S^2$.

Motivation & Objective

• To establish the existence of periodic choreography solutions for the $n$-body problem on the unit sphere $S^2$ with positive curvature.
• To extend the classical planar eight-shape choreography solution (discovered by Chenciner and Montgomery) to the curved geometry of $S^2$.
• To prove that the Lagrangian action functional achieves its minimum on a symmetric subset $H$ corresponding to choreographic motions.
• To show that this minimum corresponds to a non-collision, periodic solution of the equations of motion on $S^2$.
• To provide a variational framework for constructing new families of symmetric periodic solutions in curved celestial mechanics.

Proposed method

• Define the Lagrangian action functional $f(q) = \int_0^1 \left(\frac{1}{2}\sum m_i|\dot{q}_i|^2 + U(q)\right) dt$ on the configuration space $D$ of absolutely continuous, non-colliding $n$-tuples on $S^2$.
• Introduce symmetric subspaces $E_1$, $E_2$, $E_3$ corresponding to cyclic shift, time-reversal, and spatial reflection symmetries, and define $H = E_1 \cap E_2 \cap E_3$ as the set of choreographic solutions.
• Apply Palais' principle of symmetric criticality to ensure that critical points of $f$ restricted to $H$ are also critical points in the full space $D$, hence solutions of the equations of motion.
• Use the potential function $U = \sum_{i<j} m_i m_j \cot(d(q_i,q_j))$ to model gravitational interaction on $S^2$, generalizing Newtonian gravity to spherical geometry.
• Estimate the action lower bound for binary collision solutions using inequalities involving Euclidean distance $r_{ij}$ and $\cot(d(q_i,q_j))$, proving that the true minimum lies above the collision threshold.
• Construct a test loop with $q_i(t) = q_1(t + i/3)$ and specific parametric coordinates to numerically verify that the action is below the collision threshold, confirming the existence of a non-collision solution.

Experimental results

Research questions

• RQ1 Does a non-collision choreography solution exist for the three-body problem on the 2-sphere ($S^2$) with positive curvature?
• RQ2 Can the classical planar eight-shape choreography be generalized to spherical geometry via variational methods?
• RQ3 Is the minimum of the Lagrangian action functional on the symmetric subset $H$ achieved at a regular, non-colliding solution?
• RQ4 What is the lower bound of the action functional for generalized binary collision solutions on $S^2$, and how does it compare to the actual action of a test loop?
• RQ5 Can symmetric criticality be applied effectively to prove existence of periodic solutions in the curved $n$-body problem?

Key findings

• The Lagrangian action functional reaches its minimum on the symmetric subset $H$, which corresponds to choreographic motions on $S^2$, and this minimum is a non-collision, periodic solution of the equations of motion.
• The minimum action is strictly less than $\frac{3}{2}(12\pi)^{2/3} - 3 \approx 13.8647$, and a test loop with action $f(q) \approx 13.76572$ confirms that the infimum is not achieved at a collision solution.
• The paper proves that the action of any binary collision solution is bounded below by $\frac{3}{2}(12\pi)^{2/3} - 3$, and since the test loop has lower action, the true minimizer must be a regular solution.
• The 8-shaped orbit on $S^2$ is realized as a non-colliding, symmetric periodic solution with three equal masses, satisfying cyclic, time-reversal, and reflection symmetries.
• The existence of such a solution is established via variational methods and symmetric criticality, extending the classical planar choreography to curved space.
• The potential function $U = \sum_{i<j} \cot(d(q_i,q_j))$ correctly generalizes Newtonian gravity to $S^2$, and the inequality $\frac{1}{r_{ij}} - 1 < \cot(d(q_i,q_j)) < \frac{1}{r_{ij}}$ is used to bound the action from below.