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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 $W:={V M$ 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 $(h-V\circ\tau_M^\ast)^{-1}K$.

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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:

• $W:={V •$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\$.