Action-Angle Variables

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

Action-Angle Variables are standard tools for analyzing integrable Hamiltonian systems, consisting of a pair of conjugate variables $(I, \theta)$ :

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

For a one-dimensional integrable Hamiltonian system, the introduction of action-angle variables makes the Hamiltonian depend only on the action $H = H(I)$ , and the conjugate equation for the angle variable gives a constant angular velocity $\dot{\theta} = \partial H / \partial I$ .

Significance in Hamiltonian Systems

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

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

The integration is performed over one complete circuit along the closed orbit. As the angle variable $\theta$ varies from $0$ to $2\pi$ , the action $I$ remains constant.

After introducing action-angle variables, the phase space geometry of an integrable Hamiltonian system becomes extremely clear: $I$ describes the radial section of an invariant torus, and $\theta$ describes the rotational phase of the orbit on the torus.

Definition in Linearized Dynamics at Libration Points

The linearized Hamiltonian of the CR3BP collinear libration points, after Birkhoff-Gustavson normalization, has a saddle × center × center structure. For the center modes, Qiao et al. (2025) adopt the following action-angle variable definitions:

Center mode:

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

Saddle mode:

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

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

Application in Orbit Characteristic Parameters

Qiao et al. (2025) ultimately select a six-dimensional characteristic parameter set:

Parameter	Type	Definition	Physical Meaning
$q_1$	Saddle coordinate	Retained directly (no action-angle transformation)	Degree of entry along the unstable manifold
$p_1$	Saddle momentum	Retained directly (no action-angle transformation)	Degree of entry along the 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 angle variable definition for the saddle involves complex variables, with abstract physical meaning, making practical application inconvenient.
The numerical values of $q_1, p_1$ themselves are sufficient to describe the spacecraft's degree of entry into the unstable/stable manifold ( $q_1$ growing exponentially indicates moving away along the unstable manifold, $p_1$ decaying exponentially to zero indicates approaching along the stable manifold).

Equations of Motion on the Central Manifold

After introducing 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)

From this, the canonical equations are derived:

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 under action-angle variables, the motion on the central manifold has a clear integrable structure (linear part) + perturbation correction.

Central Manifold
Birkhoff-Gustavson Normal Form
Poincare Section
Canonical Transformation
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.