classical-mechanics

In Lagrangian formalism, if $(M, g)$ is our configuration manifold, equipped with a Riemannian metric $g\in Hom(TM\otimes TM, \mathbb{R})$ , Lagrangian function $\mathcal{L}: TM\times \mathbb{R}_{+}\rightarrow \mathbb{R}$ is defined as $$\mathcal{L}(x(t), \dot{x}(t), t)= \frac{1}{2}mg(\dot{x}(t), \dot{x}(t))-U(x(t), t).$$ If our particle flows under the influence of gravity, then $$\frac{d}{dt}\b…