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Lagrange Planetary Equations

A set of differential equations describing the time rate of change of the orbital elements (e.g., semi-major axis aa, eccentricity ee, inclination ii) due to perturbing forces F⃗\vec{F}. They are fundamental for numerical integration.
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The statement of the theorem

Let L=(a,e,i,Ω,ω,M)\mathbf{L} = (a, e, i, \Omega, \omega, M) be the set of classical orbital elements, and F⃗\vec{F} be the perturbing force per unit mass. The time rate of change of these elements is given by:\ndLdt=∂θ∂elements⋅F⃗r2×r×h\frac{d\mathbf{L}}{dt} = \frac{\partial \boldsymbol{\theta}}{\partial \text{elements}} \cdot \frac{\vec{F}}{r^2} \times \mathbf{r} \times \mathbf{h} \nSpecifically, the equations for the rate of change of the semi-major axis aa and eccentricity ee are:\na˙=1−e2na2e[(1−e2)F⃗⋅r+er2r˙F⃗⋅t]\dot{a} = \frac{\sqrt{1-e^2}}{n a^2 e} [ (1-e^2) \vec{F} \cdot \mathbf{r} + e r^2 \dot{r} \vec{F} \cdot \mathbf{t} ] \ne˙=1na2e[(1−e2)F⃗⋅r+er2r˙F⃗⋅t]\dot{e} = \frac{1}{n a^2 e} [ (1-e^2) \vec{F} \cdot \mathbf{r} + e r^2 \dot{r} \vec{F} \cdot \mathbf{t} ] \n(where nn is the mean motion, r\mathbf{r} is the radial unit vector, and t\mathbf{t} is the transverse unit vector).
Source: Wikipedia