Transverse foliations in the rotating Kepler problem

Abstract

We construct finite energy foliations and transverse foliations of neighbourhoods of the circular orbits in the rotating Kepler problem for all negative energies. This paper would be a first step towards our ultimate goal that is to recover and refine McGehee's results on homoclinics and to establish a theoretical foundation to the numerical demonstration of the existence of a homoclinic-heteroclinic chain in the planar circular restricted three-body problem, using pseudoholomorphic curves.

Figures (14)

• Figure 1: Hill's regions corresponding to energies slightly above $H_\mu(L_1)$ (left) and $H_\mu(L_2)$ (right). The darkly shaded regions indicate the projection of invariant tori.
• Figure 2: A homoclinic-heteroclinic chain in the PCR3BP. The inner and outer black curves denote homoclinic orbits to the Lyapunoff orbits (red curves) near $L_1$ and $L_2$, respectively. The blue curve indicates a heteroclinic orbit.
• Figure 3: The solution set of (left) $2E(c-E)^2+1=0$; (right) $2(c+L)L^2+1=0$ is the union of two graphs. The left and right curves indicate the retrograde and direct circular orbits, respectively.
• Figure 4: Some $T_{k,l}$-tori with $k=9.$ The dashed line indicates the tori consisting only of collision orbits.
• Figure 5: Some quantities associated with a Kepler ellipse of Kepler energy $E<0$. The quantity $a$ is the semi-major axis that is equal to $-\frac{1}{2E}.$ The quantity $\alpha$ is given by the argument of the present position, and $\beta$ is given by the argument of the perihelion.

Theorems & Definitions (20)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• Definition 2.2
• Theorem 2.3: Hofer, Wysocki and Zehnder HWZ03foliation
• Lemma 3.1
• Theorem 4.1: Lagrange, 1808
• Remark 4.2
• Remark 5.1
• Lemma 5.2