Starting from Kevin Lin's answer I find that contact geometry gives a proof of the Maupertuis principle which is geometric and doesn't appeal at the variational principle of Least Action.

Let be given an Hamiltonian system with configuration space $M$, potential energy $V$ and kinetic energy $K$.

For any regular value $h$ of the Hamilton function $H:=K+V\circ\tau_M^\ast$, let us introduce:

- W the open subset of $M$ where $h-V$ is positive, and
- $N:=H^{-1}(h)\setminus K^{-1}(0)$, a codimension-1 submanifold of $T^\ast W$.

Let us define $\tilde{K}=(h-V\circ\tau_M^\ast)^{-1}K|_{T^\ast W}$, a metric on $W$, seen as a smooth function on $T^\ast W$, which is a fiberwise positive definite quadratic form.

Let us denote by $X$ and $\tilde{X}$ the Hamiltonian vector fields on $(T^\ast W,d\lambda)$ havind as Hamilton functions $H$ and $\tilde{K}$ respectively.

By definition, $N:=H^{-1}(h)\setminus K^{-1}(0)$ coincides with $\tilde{K}^{-1}(1)$, and the Liouville $1$-form $\lambda$ induces a contact form on it.

From the following pair of identities:

- $i(X)\lambda=2K$, $i(X)d\lambda=-dH$, and
- $i(\tilde{X})\lambda=2\tilde{K}$, $i(\tilde{X})d\lambda=-d\tilde{K}$,

we deduce that:

- $X$ and $\tilde{X}$ are tangent to $N$,
- both $(2K)^{-1}X|_N$ and $2\tilde{K})^{-1}\tilde{X}|_N\equiv 1/2\tilde{X}|_N$ satisfy the defining equations for the Reeb vector field on the strictly contact manifold $(N,j_N^\ast\lambda)$.

So on $N:=H^{-1}(h)\setminus K^{-1}(0)$, the Hamiltonian vector field $X$ of $H:=K+V\circ\tau_M^ast$ coincides with $2K\tilde{X}$, being $\tilde{X}$ the geodesic vector field for the Jacobi metric on $W$ given by $(h-V\circ\tau_M^\ast)^{-1}K$.