Lagrangian Perturbation Equations

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

Definition

The Lagrangian perturbation equations are disturbed equations of motion expressed in terms of orbital elements, first established by Lagrange during his study of planetary perturbation motion. These equations represent the rate of change of each orbital element as a function of the six orbital elements and the partial derivatives of the disturbing function with respect to each orbital element. They are applicable only to conservative perturbation forces (such as non-spherical Earth gravity and lunar-solar gravitational attraction).

Core Elements

Equation Form

\left\{\begin{array}{l} \dot{a} = \frac{2}{na}\frac{\partial R}{\partial M} \\ \dot{e} = \frac{1-e^2}{na^2e}\frac{\partial R}{\partial M} - \frac{\sqrt{1-e^2}}{na^2e}\frac{\partial R}{\partial\omega} \\ \dot{i} = \frac{1}{na^2\sqrt{1-e^2}\sin i}(\cos i\frac{\partial R}{\partial\omega} - \frac{\partial R}{\partial\Omega}) \\ \dot{\Omega} = \frac{1}{na^2\sqrt{1-e^2}\sin i}\frac{\partial R}{\partial i} \\ \dot{\omega} = \frac{\sqrt{1-e^2}}{na^2e}\frac{\partial R}{\partial e} - \cos i\frac{d\Omega}{dt} \\ \dot{M} = n - \frac{1-e^2}{na^2e}\frac{\partial R}{\partial e} - \frac{2}{na}\frac{\partial R}{\partial a} \end{array}\right.

Symmetry Properties

Orbital Element Group	Rate of Change Depends On
First three: $\dot{a}$ , $\dot{e}$ , $\dot{i}$	Only $\partial R/\partial\Omega$ , $\partial R/\partial\omega$ , $\partial R/\partial M$
Last three: $\dot{\Omega}$ , $\dot{\omega}$ , $\dot{M}$	Only $\partial R/\partial a$ , $\partial R/\partial e$ , $\partial R/\partial i$

The rate of change of any group of orbital elements depends only on the partial derivatives of the disturbing function with respect to the other group. This property is known as "symmetry."

Comparison with Gaussian Equations

Property	Gaussian	Lagrangian
Applicable perturbation forces	Arbitrary (conservative + non-conservative)	Conservative only
Input quantities	Three perturbation acceleration components	Partial derivatives of disturbing function
Physical insight	Directly reflects force action	Reveals relationship between disturbing function and orbital changes
Primary use	Atmospheric drag, thrust analysis	Non-spherical Earth, lunar-solar gravity analysis

Derivation Approach

Starting from the Gaussian equations and utilizing the mapping relationships between $\partial R/\partial\sigma$ and perturbation force components $f_r$ , $f_u$ , $f_h$ , the Lagrangian equations are derived through inverse transformation. The key step is expressing the partial derivatives of the disturbing function with respect to orbital elements as functions of the perturbation force components.

Application Value

The Lagrangian perturbation equations are the standard tool for analyzing the effects of conservative perturbation forces (especially non-spherical Earth gravity) on orbits. By substituting the Earth's oblateness disturbing function into the equations, the influence of $J_2$ secular terms on each orbital element can be derived, providing the theoretical foundation for the design of special orbits such as sun-synchronous orbits and frozen orbits.

References

郑伟, 安雪滢, 周祥, 何睿智. 空天飞行力学[M]. 国防科技大学, 2026.
刘林. 航天器轨道理论[M]. 国防工业出版社.