Central Manifold

Author: Tianjiang Talk
Edited from: Qiao et al. (2025) "Orbital parameter characterization and objects cataloging for Earth-Moon collinear libration points"
Website: https://cislunarspace.cn

Definition

The Central Manifold is the invariant manifold composed of the neutral stable directions in the phase space decomposition near an equilibrium point of a Hamiltonian system. In the dynamics study of cislunar libration points (especially collinear libration points $L_1, L_2, L_3$ ), central manifold theory is the core tool for dimension reduction of high-dimensional phase space and revealing orbital geometric structure.

In the neighborhood of CR3BP collinear libration points, the linearized dynamics exhibit a saddle × center × center structure:

Hyperbolic directions (saddle/unstable): along unstable and stable manifolds, state variables grow or decay exponentially
Center directions (center/stable): the two center modes move along elliptical orbits, with bounded oscillatory state variables

The central manifold corresponds to the collection of both center directions, forming the low-dimensional invariant manifold on which libration point orbit families (such as Halo orbits, Lyapunov orbits, and Lissajous orbits) reside.

Dynamics Structure

For the linearized Hamiltonian of CR3BP collinear libration points:

H_2 = \lambda q_1 p_1 + \frac{\omega_p}{2}(q_2^2 + p_2^2) + \frac{\omega_\nu}{2}(q_3^2 + p_3^2)

The three eigenvalues correspond to the following motion patterns:

Eigenvalue	Symbol	Motion Character
$\lambda$	Square root of $\eta_1 < 0$	Hyperbolic: $q_1$ grows exponentially, $p_1$ decays exponentially
$\omega_p$	Square root of $\eta_2 > 0$	Center: elliptical motion in the $XY$ plane
$\omega_\nu$	Square root of $\eta_3 > 0$	Center: oscillatory motion in the $Z$ direction

Decoupling from Hyperbolic Invariant Manifolds

One of the core contributions of Qiao et al. (2025) is the decoupling of the hyperbolic directions from the central manifold via canonical transformation. The decoupled Hamiltonian takes the form:

\hat{H} = H_2 + \hat{H}_N(q_1 p_1, I_2, I_3, \theta_2, \theta_3) + R_N(q,p)

where $R_N$ is the high-order remainder beyond order $N$ . The key effect of this decoupling is:

The hyperbolic directions $q_1, p_1$ appear in the higher-order terms only as the product form $q_1 p_1$ , no longer coupled with the center direction coordinates
Motion on the central manifold is described by only four parameters $(I_2, \theta_2, I_3, \theta_3)$

This decoupling enables independent treatment of hyperbolic escape/capture dynamics (stable/unstable manifolds) and periodic/quasi-periodic orbital motion when studying libration point orbit families.

Motion on the Central Manifold

On the central manifold, motion can be described by action-angle variables. Let the central manifold Hamiltonian be:

H_{CM} = H_2(I_2, I_3) + H_N(I_2, I_3, \theta_2, \theta_3)

Then:

Actions $I_2, I_3$ are constant in the linear part (integrals), with long-period perturbation oscillations arising from higher-order terms
Angle variables $\theta_2, \theta_3$ vary linearly: $\theta = \omega t + \theta_0$ , with small-amplitude vibrations superimposed from higher-order terms

Poincare Section Analysis

Motion on the central manifold is a two-dimensional torus motion in a four-dimensional phase space. Using a Poincare section ( $\theta_2 = 0$ or $\theta_3 = \pi/2$ ) reduces the phase space dimension for visualization.

On the Poincare section:

Lyapunov orbits: section intersection curves corresponding to $I_3 = 0$
Vertical Lyapunov orbits: section intersection curves corresponding to $I_2 = 0$
Halo orbits: the torus contracts to the central point, corresponding to specific energy layers near $\theta_3 = \pi/2$ or $3\pi/2$
Lissajous orbits: ergodic trajectories on the section, where $\theta_3$ traverses the entire section in $[0, 2\pi)$
Quasi-Halo orbits: locally forming torus structures

Definition of Invariant Manifolds

Based on the decoupled characteristic parameters, the stable manifold $W_s$ and unstable manifold $W_u$ can be succinctly defined as:

W_s = \{[q_1, p_1, I_2, \theta_2, I_3, \theta_3] \mid H_{CM} = C, \ q_1 = 0\}

W_u = \{[q_1, p_1, I_2, \theta_2, I_3, \theta_3] \mid H_{CM} = C, \ p_1 = 0\}

When $q_1 = 0$ , $p_1 \neq 0$ : $p_1$ decays exponentially to zero over time → stable manifold (approaching the libration point)
When $p_1 = 0$ , $q_1 \neq 0$ : $q_1$ grows exponentially over time → unstable manifold (departing from the libration point)

Role in Orbit Parameterization

Qiao et al. (2025) used central manifold theory to establish a six-dimensional characteristic parameter system for the neighborhood of Earth-Moon collinear libration points:

Parameter	Description	Physical Meaning
$q_1$	Hyperbolic direction coordinate	Degree of entry along the unstable manifold
$p_1$	Hyperbolic direction momentum	Degree of entry along the stable manifold
$I_2$	Central manifold action 1	Amplitude of motion in the $XY$ plane
$\theta_2$	Central manifold angle 1	Phase in the $XY$ plane
$I_3$	Central manifold action 2	Amplitude of motion in the $Z$ direction
$\theta_3$	Central manifold angle 2	Phase in the $Z$ direction

Birkhoff-Gustavson Normal Form
Poincare Section
Action-Angle Variables
Circular Restricted Three-Body Problem (CR3BP)
Hyperbolic Invariant Manifold
Stable/Unstable Manifold
Halo Orbit (Lissajous Orbit)

References

Arnol'd V I. Mathematical methods of classical mechanics[M]. Springer, 1989.
Jorba A, Masdemont J. Dynamics in the center manifold of the collinear points of the restricted three body problem[J]. Phys D, 1999, 132(1-2): 189-213.
Qiao C, Long X, Yang L, et al. Orbital parameter characterization and objects cataloging for Earth-Moon collinear libration points[J]. Chinese Journal of Aeronautics, 2025. doi: 10.1016/j.cja.2025.103869.