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The dynamics around the collinear points of the elliptic three-body problem: A normal form approach

Alessandra Celletti, Christoph Lhotka, Giuseppe Pucacco

TL;DR

This work constructs a resonant normal-form framework for the elliptic planar restricted three-body problem in a rotating-pulsating frame with the true anomaly as the independent variable. By extending the phase space and performing center-manifold reduction, the authors obtain an integrable 3-DOF reduced dynamics that captures planar and vertical Lyapunov orbits as well as halo orbits near the collinear points, including explicit expressions for the normal form and generating functions. The theory yields bifurcation thresholds and analytic approximations for orbit families, which are validated against high-precision numerical simulations and refined initial conditions, with applications illustrated for the Earth–Moon system. The approach provides a systematic, implementable path to compute invariant manifolds and resonant orbits in the elliptic problem, offering improvements over previous methods and enabling practical trajectory design in astrodynamics.

Abstract

We study the dynamics of the collinear points in the planar, restricted three-body problem, assuming that the primaries move on an elliptic orbit around a common barycenter. The equations of motion can be conveniently written in a rotating pulsating barycentric frame, taking the true anomaly as independent variable. We consider the Hamiltonian modeling this problem in the extended phase space and we imple ment a normal form to make a center manifold reduction. The normal form provides an approximate solution for the Cartesian coordinates, which allows us to construct several kinds of orbits, most notably planar and vertical Lyapunov orbits, and halo orbits. We compare the analytical results with a numerical simulation, which requires special care in the selection of the initial conditions.

The dynamics around the collinear points of the elliptic three-body problem: A normal form approach

TL;DR

This work constructs a resonant normal-form framework for the elliptic planar restricted three-body problem in a rotating-pulsating frame with the true anomaly as the independent variable. By extending the phase space and performing center-manifold reduction, the authors obtain an integrable 3-DOF reduced dynamics that captures planar and vertical Lyapunov orbits as well as halo orbits near the collinear points, including explicit expressions for the normal form and generating functions. The theory yields bifurcation thresholds and analytic approximations for orbit families, which are validated against high-precision numerical simulations and refined initial conditions, with applications illustrated for the Earth–Moon system. The approach provides a systematic, implementable path to compute invariant manifolds and resonant orbits in the elliptic problem, offering improvements over previous methods and enabling practical trajectory design in astrodynamics.

Abstract

We study the dynamics of the collinear points in the planar, restricted three-body problem, assuming that the primaries move on an elliptic orbit around a common barycenter. The equations of motion can be conveniently written in a rotating pulsating barycentric frame, taking the true anomaly as independent variable. We consider the Hamiltonian modeling this problem in the extended phase space and we imple ment a normal form to make a center manifold reduction. The normal form provides an approximate solution for the Cartesian coordinates, which allows us to construct several kinds of orbits, most notably planar and vertical Lyapunov orbits, and halo orbits. We compare the analytical results with a numerical simulation, which requires special care in the selection of the initial conditions.
Paper Structure (15 sections, 1 theorem, 83 equations, 8 figures)

This paper contains 15 sections, 1 theorem, 83 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Collinear points and center manifold reduction
  3. The rotating-pulsating Hamiltonian
  4. The collinear points
  5. The Hamiltonian in the extended phase space
  6. The quadratic part of the Hamiltonian
  7. Center manifold reduction
  8. The resonant normal form
  9. Halo orbits
  10. Example: $L_1$ in the Earth-Moon case
  11. Normal-mode frequencies
  12. Planar Lyapunov orbits
  13. Vertical Lyapunov orbits
  14. Halo orbits
  15. Validation by numerical simulations

Key Result

Proposition 1

Consider the Hamiltonian ${\mathcal{H}} (p,q,J_4,f)$ in (Hc). There exists a canonical transformation ${\mathcal{C}}:{\mathbb R}^6\rightarrow{\mathbb R}^6$ with ${\mathcal{C}}(p,q)=(P,Q)$, such that ${\mathcal{H}}$ is transformed to order $N\in{\mathbb N}$ into the 'normal form' where the polynomials $K_{n}$ depend on $Q_1,P_1$ only through their product $Q_1P_1$, while $R_{N+1}(P,Q)$ is the rema

Figures (8)

  • Figure 1: $\Omega_y$ (blue) and $\Omega_z$ (yellow) for the point $L_1$ as a function of the eccentricity in the Earth-Moon system, computed using a normal form to order 2 and a series expansion of the eccentricity to order 8.
  • Figure 2: A planar Lyapunov orbit around $L_1$ in the Earth-Moon ($\mu=0.012$) system with ${\rm e}=0.2$. Units are in terms of the Earth-Moon distance.
  • Figure 3: Periodic planar Lyapunov orbits (in yellow) for $\kappa_y=$ 2:1 in the Earth-Moon system: on the left we use ${\rm e}=0.2$ in order to magnify the effect. On the right the true value ${\rm e}=0.0549$ is used. The two orbits are respectively produced by the amplitudes $J_y=1.075$ and $J_y=0.9287$. In blue the corresponding orbit in the circular case.
  • Figure 4: Periodic vertical Lyapunov orbits for $\kappa_z=$ 2:1 in the Earth-Moon ($\mu=0.012$) system but with ${\rm e}=0.2$. The amplitude is $J_z=0.9775$.
  • Figure 5: Periodic vertical Lyapunov orbits for $\kappa_z=$ 3:2 in the Earth-Moon ($\mu=0.012$) system but with ${\rm e}=0.2$. The amplitude is $J_z=2.7030$.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Proposition 1