[Paper Review] Nonplanar Periodic Solutions for Spatial Restricted N+1-Body Problems

Fengying Li, Shiqing Zhang+1 more|arXiv (Cornell University)|Sep 6, 2012

Spacecraft Dynamics and Control7 references3 citations

TL;DR

This paper establishes the existence of nonplanar periodic solutions in spatial restricted N+1-body problems using variational minimization methods. By minimizing the Lagrangian action on anti-symmetric or odd-symmetric loop spaces, it proves that the minimizers for N ≥ 2 are non-colliding, nonplanar periodic orbits confined to the vertical axis, avoiding the central plane due to Jacobi's necessary condition violation at the origin.

ABSTRACT

We use variational minimizing methods to study spatial restricted N+1-body problems with a zero mass moving on the vertical axis of the moving plane for N equal masses. We prove that the minimizer of the Lagrangian action on the anti-T/2 or odd symmetric loop space must be a non-planar periodic solution for any $N\geq2$.

Motivation & Objective

• To establish the existence of nonplanar periodic solutions in spatial restricted N+1-body problems with N equal masses in circular motion.
• To analyze the dynamics of a zero-mass particle constrained to move along the vertical axis above a rotating N-body configuration.
• To prove that minimizers of the Lagrangian action on symmetric loop spaces yield non-colliding, nonplanar periodic solutions for N ≥ 2.
• To apply variational methods and Jacobi's necessary condition to rule out planar solutions centered at the origin.
• To demonstrate that the minimizer cannot be at rest at the center of mass, implying oscillatory, nonplanar motion.

Proposed method

• Formulates the problem using a Lagrangian action functional f(q) = ∫₀¹ [½|ż|² + N/√(r² + z²)] dt for vertical motion along the z-axis.
• Imposes symmetry constraints: Λ₁ (anti-T/2 symmetry) and Λ₂ (odd symmetry) to restrict the search space for periodic solutions.
• Applies Palais’s Symmetry Principle to ensure critical points in symmetric subspaces are critical in the full space.
• Uses coercivity and weak lower semicontinuity in W¹,²(R/Z,R) to guarantee attainment of the infimum of f(q) on the closure of Λᵢ.
• Applies the Poincaré-Wirtinger inequality to control the L² norm of the derivative and ensure compactness.
• Employs Jacobi’s necessary condition by analyzing the second variation of the action functional around z=0, showing conjugate points exist.

Experimental results

Research questions

• RQ1 Can nonplanar periodic solutions exist in the spatial restricted N+1-body problem for N ≥ 2?
• RQ2 Is the minimizer of the Lagrangian action on symmetric loop spaces (Λ₁ or Λ₂) necessarily nonplanar?
• RQ3 Does the absence of a local minimum at z=0 imply that the minimizer must oscillate off the central plane?
• RQ4 What is the role of Jacobi’s necessary condition in ruling out planar solutions?
• RQ5 How does the radius r of the N-equal-mass orbit affect the existence and nature of nonplanar solutions?

Key findings

• The minimizer of the Lagrangian action on the closure of Λ₁ or Λ₂ is a non-collision, nonplanar periodic solution for any N ≥ 2.
• The functional f(q) attains its infimum on both Λ₁ and Λ₂ due to weak lower semicontinuity and coercivity in the Sobolev space W¹,².
• Jacobi’s necessary condition fails at z=0, as the solution to the Jacobi equation h'' + (N/r³)h = 0 has a conjugate point at t = 1/2 and another at t = 1/(2N) < 1/2.
• The radius r of the N-mass orbit is given by r = (1/(4π))^(2/3) [∑_{j=1}^{N-1} csc(πj/N)]^(1/3), derived from Newtonian equations of motion.
• The existence of conjugate points in (0, ½) implies that z ≡ 0 is not a local minimizer, so the minimizer must oscillate vertically, breaking planarity.
• The minimizer cannot be co-planar with the N masses, confirming the solution is strictly nonplanar.