Central Manifold

Source: 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 neutral-stable invariant manifold formed by the center directions in the phase space decomposition near a Hamiltonian system equilibrium point. In cislunar space libration point (particularly collinear libration points $L_1, L_2, L_3$ ) dynamics, central manifold theory is the core tool for dimensionality reduction and revealing orbital geometric structure.

In the linearized CR3BP near collinear libration points, the dynamics have a saddle × center × center structure:

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

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

Dynamical Structure

For the linearized CR3BP Hamiltonian near a collinear libration point:

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 eigenmodes correspond to:

Eigenvalue	Symbol	Motion Character
$\lambda$	$\sqrt{-\eta_1}$	Hyperbolic: $q_1$ grows exponentially, $p_1$ decays exponentially
$\omega_p$	$\sqrt{\eta_2}$	Center: elliptical motion in the $XY$ plane
$\omega_\nu$	$\sqrt{\eta_3}$	Center: oscillatory motion in the $Z$ direction

Decoupling from Hyperbolic Invariant Manifold

A core contribution of Qiao et al. (2025) is the decoupling of hyperbolic directions from the central manifold via canonical transformation. After decoupling, the 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 remainder beyond order $N$ . The key effect of decoupling is:

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

This decoupling allows independent treatment of hyperbolic escape/capture dynamics (stable/unstable manifolds) and periodic/quasi-periodic orbital motion.

Motion on the Central Manifold

On the central manifold, motion is described by action-angle variables. With the central manifold Hamiltonian:

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

we have:

Action variables $I_2, I_3$ are constant in the linear part (integrals), with higher-order terms introducing long-period oscillatory perturbations
Angle variables $\theta_2, \theta_3$ vary linearly: $\theta = \omega t + \theta_0$ , with higher-order terms adding small-amplitude vibrations

Poincaré Section Analysis

Motion on the central manifold is a 2D torus in 4D phase space. Using a Poincaré section ( $\theta_2 = 0$ or $\theta_3 = \pi/2$ ) reduces the phase space dimensionality for visualization.

On the Poincaré section:

Lyapunov orbits: intersection curve with $I_3 = 0$
Vertical Lyapunov orbits: intersection curve with $I_2 = 0$
Halo orbits: torus shrinks to a central point at specific energy levels, corresponding to $\theta_3 = \pi/2$ or $3\pi/2$
Lissajous orbits: section trajectory fills the entire available region as $\theta_3$ varies over $[0, 2\pi)$
Quasi-Halo orbits: locally form toroidal structures

Stable and Unstable Manifolds

Based on the decoupled characteristic parameters, the stable manifold $W_s$ and unstable manifold $W_u$ are 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) leverage central manifold theory to establish a six-dimensional characteristic parameter system for the collinear libration point region:

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

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

References

Arnol'd V I. Mathematical methods of classical mechanics[M]. Springer, 1989.
Jorba À, 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.