On the Formulation of Homotopical Mechanics

Let M$_{\mathrm{G}}$ be a path-connected and simply connected space over a gravitational field G. Let $\alpha_{\mathrm{i}}$ be a free-fall path on M$_{\mathrm{G}}$ for $i\in Z$(set of integers). The Hamiltonian ${\rm H}$ for $\alpha_{\mathrm{i}}$ obeys the homotopy theory. We showed that the Euler-Lagrange's equations for $\alpha_{\mathrm{i}}$ can be expressed in terms of a homotopy map between the kinetic energy u(t) and potential energy v(t) for $\alpha_{\mathrm{i}}$, here t$\in $ [0,1].