Is there an action functional whose critical points are the geodesics of a arbitrary connection on TM?

Question

Geodesics of the Levi-Civita connection may be defined as the critical points of the action functional $S[\gamma]=\int \lvert\dot{\gamma}\rvert\,dt$ (or square it, if you like). The Euler-Lagrange equation of motion for this action is $\nabla_{\dot{\gamma}}\dot{\gamma}=0,$ where $\nabla$ is the Levi-Civita connection. This equation makes sense of course for any connection, so we may define geodesics with respect to that connection. Is there some functional on paths which depends only on the connection and yields this EOM?