Equation of Motion and State Equation

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

An equation of motion (EOM) is the differential equation describing how a spacecraft's position evolves under a specified force model. In the circular restricted three-body problem, written in the synodic (rotating) frame, the EOMs are three second-order ODEs whose right-hand side combines the gravitational pull of the two primaries with the Coriolis and centrifugal terms inherent to the rotating frame. Rewriting any system of second-order ODEs as a first-order system yields the state equation, the standard form on which modern control theory operates.

The CR3BP equations of motion

In the synodic frame, with origin at the barycentre and the x-axis along the two primaries (which are then fixed), the dimensional equations are (Szebehely 1967, §1.5; Vallado 2022, §12.3):

\ddot{\mathbf r}_S + 2\,\boldsymbol{\omega}_S \times \dot{\mathbf r}_S + \boldsymbol{\omega}_S \times (\boldsymbol{\omega}_S \times \mathbf r_S) = -\sum_{i=1,2} G m_i \frac{\mathbf r_S - \mathbf r_i}{\|\mathbf r_S - \mathbf r_i\|^3}.

The second term is the Coriolis acceleration, the third the centrifugal acceleration. After non-dimensionalisation (cf. Nondimensionalization) and introducing the effective potential

\Omega(x,y,z) = \tfrac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2},

the equations take their canonical compact form

\ddot{x} - 2\dot{y} = \Omega_x, \qquad \ddot{y} + 2\dot{x} = \Omega_y, \qquad \ddot{z} = \Omega_z.

Because time does not appear explicitly, this system is autonomous, and admits the Jacobi integral $C = 2\Omega - (\dot x^2 + \dot y^2 + \dot z^2)$ as a conserved quantity — the foundation on which zero-velocity surfaces, libration points, and all CR3BP-based mission design rest (Szebehely 1967, §1.6).

The state equation

Defining the state vector $\mathbf x = (x, y, z, \dot x, \dot y, \dot z)^\top$ , the second-order EOMs become a first-order system

\dot{\mathbf x} = \mathbf f(\mathbf x),

and, in the controlled linearised form used in modern control,

\dot{\mathbf x} = \mathbf A(t)\,\mathbf x + \mathbf B(t)\,\mathbf u, \qquad \mathbf y = \mathbf C(t)\,\mathbf x.

This is the state equation (state-space form). Casting the dynamics in first-order form is the prerequisite for applying optimal control, state feedback, and state-observer techniques. In the CR3BP, when the reference is a periodic orbit, $\mathbf A(t)$ is periodic with the orbit's period — this is the linear time-periodic structure that underpins Floquet analysis of relative motion about libration-point orbits.

Autonomous vs. non-autonomous; time-varying vs. time-invariant

A system is autonomous (time-invariant) when $\mathbf f$ does not depend explicitly on $t$ ; otherwise it is non-autonomous (time-varying). The CR3BP state equation is autonomous because both primaries are stationary in the synodic frame. The bi-circular problem, the elliptic restricted three-body problem, and the full ephemeris (N-body) model all introduce explicit time dependence through the moving third body or the real planetary positions; they are non-autonomous, lose the Jacobi integral, and require quasi-periodic or entirely numerical methods (Baresi 2023).

Autonomous systems are invariant under time shifts: a trajectory launched at $t_0$ and one launched at $t_0 + \Delta t$ have the same shape. Non-autonomous systems lose this symmetry — launch epoch matters, and numerical integration must carry the absolute time along with the state. The added difficulty is not cosmetic: the entire apparatus of Poincaré sections, invariant manifolds, and Jacobi-constrained transit-orbit theory relies on autonomy and is unavailable in the ephemeris model without modification.

References

Szebehely, 1967, Theory of Orbits, §§1.5–1.6 — dimensional and dimensionless CR3BP equations, derivation of the Jacobi integral.
Vallado, 2022, Fundamentals of Astrodynamics and Applications, §12.3 — restricted three-body problem and synodic-frame equations.
Baresi, 2023, "Transition of two-dimensional quasi-periodic invariant tori in the real-ephemeris model of the Earth–Moon system" — non-autonomous dynamics in the full ephemeris model.
Fossà et al., 2022, "Two- and three-impulse phasing strategy with a spacecraft orbiting an Earth–Moon NRHO."
Xu Ming & Xu Shijie, 2008, "Linear periodic station-keeping control strategy for halo orbits" — state equation and periodic $\mathbf A(t)$ for halo station-keeping.