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Formulating the Lagrangian

The kinetic energy we use when studying non-constant gravity is the same as before. Consequently, Equation (??) is also true for this Lagrangian.

Potential energy

If we cannot assume constant gravity, then: V=GMm(rbase+h(s))−GMmrsummit.\begin{aligned} V &=\dfrac{GMm}{(r_{\mathrm{base}} + h(s))} - \dfrac{GMm}{r_{\mathrm{summit}}}. \end{aligned} Factorizing: V=GMm((rbase+h(s))−1−rsummit−1).\begin{aligned} V &= GMm\left((r_{\mathrm{base}} + h(s))^{-1} - r_{\mathrm{summit}}^{-1}\right). \end{aligned} Where GG is the gravitational constant and MM is the mass of the Earth.

Substituting Equation (??) into Equation (2) yields:

V=GMm[(rsummit−∫0ssin⁡θ(σ)dσ)−1−rsummit−1].\begin{aligned} V &= GMm\left[\left(r_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right)^{-1}-r_{\mathrm{summit}}^{-1}\right]. \end{aligned}

Final Lagrangian

The Lagrangian of our system is therefore:

L=ms˙22−GMm[(rsummit−∫0ssin⁡θ(σ)dσ)−1−rsummit−1].\begin{aligned} L &= \dfrac{m\dot{s}^2}{2} - GMm\left[\left(r_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right)^{-1}-r_{\mathrm{summit}}^{-1}\right]. \end{aligned}

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Last modified: October 23, 2026.