[Paper Review] A convex embedding for the rotating Kepler problem

Urs Frauenfelder, Otto van Koert+1 more|arXiv (Cornell University)|May 23, 2016

Spacecraft Dynamics and Control4 references3 citations

TL;DR

This paper constructs a convex symplectic embedding of the bounded energy hypersurfaces of the planar rotating Kepler problem into $ℝ^4$ by composing Ligon-Schaaf and Levi-Civita regularization maps. The key result is that the Gauss-Kronecker curvature of the embedded hypersurface is positive, proving dynamical convexity and opening new avenues for attacking the Birkhoff conjecture via holomorphic curve methods.

ABSTRACT

In this note, we prove that below the first critical energy level, a proper combination of the Ligon-Schaaf and Levi-Civita regularization mappings provides a convex symplectic embedding of the energy surfaces of the planar rotating Kepler problem into $R^4$ endowed with its standard symplectic structure. A direct consequence is the dynamical convexity of the planar rotating Kepler problem, which has been established by Albers-Fish-Frauenfelder-van Koert by direct computations. This result opens up new approaches to attack the Birkhoff conjecture about the existence of a global surface of section in the restricted planar circular three body problem using holomorphic curve techniques.

Motivation & Objective

• To establish a convex symplectic embedding of the bounded component of energy hypersurfaces in the planar rotating Kepler problem below the first critical energy level.
• To resolve the failure of the Levi-Civita regularization alone to yield convexity, by introducing a two-step regularization procedure.
• To provide a geometric foundation for attacking the Birkhoff conjecture on the existence of a global surface of section in the restricted three-body problem.
• To demonstrate that the Gauss-Kronecker curvature of the regularized energy surface is positive, implying dynamical convexity.
• To lay the groundwork for applying holomorphic curve techniques to the restricted three-body problem by ensuring convexity in a symplectic embedding.

Proposed method

• Apply the Ligon-Schaaf regularization map to the rotating Kepler problem, transforming the energy hypersurface into a new symplectic manifold.
• Then apply the Levi-Civita regularization map to the Ligon-Schaaf-regularized system, yielding a symplectic embedding into $ℝ^4$.
• Express the energy level set in complex coordinates $(w,z) \in \u2102^2$, where the Hamiltonian becomes $L.C.^*H_r = -\frac{1}{2(\|w\|^2 + \|z\|^2)^2} + 2(w_1z_2 - w_2z_1)$.
• Define a function $F = -1 + 4(w_1z_2 - w_2z_1)(\|w\|^2 + \|z\|^2)^2 - 2c(\|w\|^2 + \|z\|^2)^2$ whose zero level set is the regularized energy surface.
• Compute the Hessian of $F$ restricted to the tangent space of the energy surface and show its determinant is positive via factorization into polynomials $f_1, f_2, f_3, f_4$, all shown to be positive under energy constraints.
• Use quaternionic identification of the gradient to construct an orthogonal tangent frame and compute the curvature determinant $DH = 524288a^6 f_1f_2f_3f_4^2$, proving strict convexity.

Experimental results

Research questions

• RQ1 Can the bounded energy hypersurfaces of the rotating Kepler problem be symplectically embedded into $\u211d^4$ as strictly convex hypersurfaces below the first critical energy level $c = -3/2$?
• RQ2 Does the composition of Ligon-Schaaf and Levi-Civita regularization yield a convex embedding where either regularization alone fails?
• RQ3 Is the Gauss-Kronecker curvature of the regularized energy surface positive for all $c < -3/2$, implying dynamical convexity?
• RQ4 Can this convex embedding technique be extended to the restricted three-body problem to support the Birkhoff conjecture on global surfaces of section?
• RQ5 What is the algebraic structure of the curvature determinant, and can it be factorized to allow curvature estimation?

Key findings

• The composition of Ligon-Schaaf and Levi-Civita regularization yields a 2-to-1 symplectic embedding of the bounded energy hypersurface $\Sigma_c^b$ into $\u211d^4$ that is strictly convex for all $c < -3/2$.
• The Gauss-Kronecker curvature of the embedded hypersurface is positive, as shown by proving the determinant of the restricted Hessian of $F$ is positive and factorizes into positive terms $f_1, f_2, f_3, f_4$.
• The curvature determinant is explicitly computed as $DH = 524288a^6 f_1f_2f_3f_4^2$, with $a = \|w\|^2 + \|z\|^2$, and all factors are strictly positive under the energy constraint $c < -3/2$.
• The function $f_3$ is shown to be positive by substituting $b = \frac{1}{4a^2} + \frac{c}{2}$ and analyzing the resulting quadratic in $c$, which remains positive for $0 < a < 1$ and $c < -3/2$.
• The embedding extends smoothly to a closed strictly convex hypersurface in $\u211d^4$ over the set of collisions, confirming the global convexity of the regularized energy surface.
• As a direct consequence, the bounded component of the energy hypersurface is dynamically convex, confirming a result previously obtained by direct computation in [2].