lagrangian formalism - Is there a Maupertuis principle for General Relativity?

The motion of a point particle in classical mechanics is given by Newton's equation, $\mathbf{F}=m\mathbf{a}$. Suppose all forces considered are conservative and we have a constant total energy $h$. Let $M$ be the configuration space of our system, $T^*M$ its cotangent bundle, $(\mathbf{q},\mathbf{p})$ the natural coordinates on $T^*M$ and $\gamma$ a curve connecting the start- and end-points of our particle's motion. Then $\int_\gamma \mathbf{p}\,\mathrm{d}\mathbf{q}$ is a viable action integral. The Maupertuis theorem states that

$$\int_\gamma\mathbf{p}\,\mathrm{d}\mathbf{q}=\sqrt{2}\int_\gamma\mathrm{d}\rho,$$ where $\mathrm{d}\rho$ is the Riemannian metric given by $$\mathrm{d}\rho=\sqrt{h-U(\mathbf{q})}\,\mathrm{d}s,$$ $\mathrm{d}s$ is the standard metric on $M$ and $U(\mathbf{q})$ the potential energy. This implies that

$$\text{Newtonian mechanics $\Longleftrightarrow$ geodesic problem of some pair $(M,\mathrm{d}\rho)$}$$

The motion of a point particle in general relativity is given by the equation

$$F=a\tag{1}$$ where $a:=\nabla_{\dot\gamma}\dot\gamma$, $\gamma$ is the path of the particle, $\nabla$ is the Levi-Civita connection of the spacetime $(\mathcal{M},g)$, and $F$ is some "force" 4-vector. $F$ can be seen as an obstruction to the geodesy of $\gamma$, since $a=0$ is just the geodesic equation. (In the same way, $\mathbf{F}$ is an obstruction to geodesy on $\mathbb{R}^n$ since $\mathbf{a}=0$ $\Leftrightarrow$ $\mathbf{x}$ is a straight line and $U(\mathbf{q})$ in the case of constrained Lagrangian mechanics.) (Proofs of the above statements of classical mechanics can be found in Arnold, V.I. Mathematical Methods of Classical Mechanics. Springer, 1989.)

Is there some pair $(\mathcal{M}',g')$ for which the geodesic problem is (1), i.e. a suitable generalization of the Maupertuis principle to relativistic mechanics in curved spacetime?