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Final Lagrangian

The Lagrangian is, by definition:

L=T−V.\begin{aligned} L &= T - V. \end{aligned}

Substituting Equations (??) and (??) into Equation (1). Therefore our Lagrangian is:

L=ms˙22−mg(hsummit−∫0ssin⁡θ(σ)dσ).\begin{aligned} L &= \dfrac{m\dot{s}^2}{2} - mg\left(h_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right). \end{aligned}
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Last modified: October 23, 2026.