Action-Angle Variables

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

Definition

Action-Angle Variables are the standard tool for analyzing integrable Hamiltonian systems. A pair of conjugate variables $(I, \theta)$ :

Action $I$ : an integral along a closed orbit, describing the "size" of the orbit
Angle $\theta$ : the phase along the periodic orbital motion, increasing linearly

For a one-dimensional integrable Hamiltonian system, after introducing action-angle variables, the Hamiltonian depends only on the action: $H = H(I)$ . The conjugate equation for the angle gives a constant angular velocity $\dot{\theta} = \partial H / \partial I$ .

Significance in Hamiltonian Systems

Action-angle variables are canonical coordinates $(q, p)$ transformed via a canonical transformation $(q, p) \to (I, \theta)$ using a generating function of the form $S(q, \theta)$ . The action is defined as:

I = \frac{1}{2\pi}\oint p \, dq

The integral is taken around one period of the closed orbit. As $\theta$ varies from $0$ to $2\pi$ , the action $I$ remains constant.

With action-angle variables, the phase space geometry of an integrable Hamiltonian system becomes transparent: $I$ describes the cross-sectional radius of the invariant torus, and $\theta$ describes the rotational phase on the torus.

Definitions in Linearized Libration Point Dynamics

After Birkhoff-Gustavson normalization, the CR3BP collinear libration point linearized Hamiltonian has a saddle × center × center structure. For the center modes, Qiao et al. (2025) use:

Center mode (center mode):

I_c = \frac{1}{2}(q^2 + p^2), \quad \theta_c = \arctan\left(\frac{q}{p}\right)

Saddle mode (hyperbolic mode):

I_s = qp, \quad \theta_s = \ln\frac{\sqrt{q}}{\sqrt{p}}

where subscript $c$ denotes center motion mode and $s$ denotes the saddle/hyperbolic motion mode.

Application in Orbital Characteristic Parameters

Qiao et al. (2025) select six characteristic parameters:

Parameter	Type	Definition	Physical Meaning
$q_1$	Saddle coordinate	Retained directly (no action-angle transform)	Degree of entry along unstable manifold
$p_1$	Saddle momentum	Retained directly (no action-angle transform)	Degree of entry along stable manifold
$I_2$	Center action 1	$I_2 = \frac{1}{2}(q_2^2 + p_2^2)$	Amplitude of motion in the $XY$ plane
$\theta_2$	Center angle 1	$\theta_2 = \arctan(q_2/p_2)$	Phase in the $XY$ plane
$I_3$	Center action 2	$I_3 = \frac{1}{2}(q_3^2 + p_3^2)$	Amplitude of motion in the $Z$ direction
$\theta_3$	Center angle 2	$\theta_3 = \arctan(q_3/p_3)$	Phase in the $Z$ direction

Why not convert $q_1, p_1$ to action-angle form?

Qiao et al. (2025) give two reasons:

The saddle angle variable definition involves complex variables; the physical meaning is abstract in practical applications
The values of $q_1, p_1$ alone fully represent the degree to which a spacecraft enters the unstable/stable manifold ( $q_1$ exponential growth indicates departure along unstable manifold; $p_1$ exponential decay toward zero indicates approach along stable manifold)

Equations of Motion on the Central Manifold

With action-angle variables, the central manifold Hamiltonian is:

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

The canonical equations are:

Action variation: $\dot{I}_j = \{I_j, H_{CM}\} = \{I_j, H_N\}$ (only higher-order terms contribute)
Angular velocity: $\dot{\theta}_j = \{\theta_j, H_{CM}\} = \omega_j + \{\theta_j, H_N\}$ (linear part + higher-order perturbation)

This shows that with action-angle variables, central manifold motion has a clear integrable structure (linear part) + perturbation correction.

Central Manifold
Birkhoff-Gustavson Normal Form
Poincaré Section
Orbit Identification
Circular Restricted Three-Body Problem (CR3BP)
Invariant Torus
Integrable System

References

Arnol'd V I. Mathematical methods of classical mechanics[M]. Springer, 1989.
Peterson L T, Scheeres D J. Local orbital elements for the circular restricted three-body problem[J]. J Guid Control Dyn, 2023, 46(12): 2275-2289.
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.