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Computing Equation (??)

The conservative force term of the Euler-Lagrange equation is therefore:

∂L∂s=GMmsin⁡θ(s)(rsummit−∫0ssin⁡θ(σ)dσ)2.\begin{aligned} \dfrac{\partial L}{\partial s} &= \dfrac{GMm\sin{\theta(s)}}{\left(r_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right)^2}. \end{aligned}

Hence the functional derivative of SS with respect to ss is therefore:

δSδs=ms¨−GMmsin⁡θ(s)(rsummit−∫0ssin⁡θ(σ)dσ)2.\begin{aligned} \dfrac{\delta S}{\delta s} &= m\ddot{s} - \dfrac{GMm\sin{\theta(s)}}{\left(r_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right)^2}. \end{aligned}

The time derivative of generalized momentum is the same as above. As for QsQ_s it is:

Qs=−GMmCrrcos⁡θ(s)(rsummit−∫0ssin⁡θ(σ)dσ)2−ρCDAs˙22.\begin{aligned} Q_s &= -\dfrac{GMmC_{rr}\cos{\theta(s)}}{\left(r_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right)^2} - \dfrac{\rho C_D A \dot{s}^2}{2}. \end{aligned}

Substituting Equations (1) and (2) into Equation (??) yields:

ms¨−GMmsin⁡θ(s)(rsummit−∫0ssin⁡θ(σ)dσ)2=−GMmCrrcos⁡θ(s)(rsummit−∫0ssin⁡θ(σ)dσ)2−ρCDAs˙22.\begin{aligned} m\ddot{s} - \dfrac{GMm\sin{\theta(s)}}{\left(r_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right)^2} &= -\dfrac{GMmC_{rr}\cos{\theta(s)}}{\left(r_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right)^2} - \dfrac{\rho C_D A\dot{s}^2}{2}. \end{aligned}

Rearranging and factorizing:

ms¨=GMm(rsummit−∫0ssin⁡θ(σ)dσ)2[sin⁡θ(s)−Crrcos⁡θ(s)]−ρCDAs˙22.\begin{aligned} m\ddot{s} &= \dfrac{GMm}{\left(r_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right)^2} \left[\sin{\theta(s)}-C_{rr}\cos{\theta(s)}\right] - \dfrac{\rho C_D A\dot{s}^2}{2}. \end{aligned}

Dividing by mm:

s¨=GM(rsummit−∫0ssin⁡θ(σ)dσ)2[sin⁡θ(s)−Crrcos⁡θ(s)]−ρCDAs˙22m.\begin{aligned} \ddot{s} &= \dfrac{GM}{\left(r_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right)^2} \left[\sin{\theta(s)}-C_{rr}\cos{\theta(s)}\right] - \dfrac{\rho C_D A\dot{s}^2}{2m}. \end{aligned}
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Last modified: October 23, 2026.