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Variation of Parameters (Gauss' Method)

A method used to determine how the orbital elements change over time when subjected to non-Keplerian forces. It involves expressing the perturbed motion in terms of the unperturbed orbital elements.
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The statement of the theorem

Let x(t)\mathbf{x}(t) be the state vector (e.g., x=(r,θ,vr,vθ)T\mathbf{x} = (r, \theta, v_r, v_{\theta})^T) and x0(t)\mathbf{x}_0(t) be the unperturbed solution. The perturbed motion x(t)\mathbf{x}(t) is expressed as:\nx(t)=x0(t)+∑k=1Nϵxk(t)\mathbf{x}(t) = \mathbf{x}_0(t) + \sum_{k=1}^{N} \epsilon \mathbf{x}_k(t) \nThe variation of parameters method determines the evolution of the orbital elements L\mathbf{L} by solving the differential equations derived from the perturbation F⃗\vec{F}: \ndLdt=M(L)⋅F⃗r2×r×h\frac{d\mathbf{L}}{dt} = \mathbf{M}(\mathbf{L}) \cdot \frac{\vec{F}}{r^2} \times \mathbf{r} \times \mathbf{h} \nwhere M(L)\mathbf{M}(\mathbf{L}) is the matrix of partial derivatives relating the perturbing force components to the rates of change of the elements.