Let $M$ be a Riemannian m-Manifold with metric tensor $g$. Let $\gamma:[0,1] \rightarrow M$ be some sufficiently regular (lets say piecewise differeniable) path in M. Is there some canonical way to let $\gamma$ "flow" (keeping $\gamma(0)$ and $\gamma(1)$ fixed) into a geodesic between $\gamma(0)$ and $\gamma(1)$? If so how much does such a flow depend on the actual parametrisation of $\gamma$?
My idea would be to use the energy functional $$E(\gamma)=\int_0^1g_{\gamma'(t)}(\gamma'(t),\gamma'(t))dt$$ and its functional derivative $\frac{\partial E}{\partial\gamma}$. As the geodesics are exactly the critical points of $E$ I would hope to find a function $$\Psi(r,t):[0,R]\times[0,1]\rightarrow M$$ such that for any $0<r\leq R$ $$\frac{\partial\Psi(r,t)}{\partial r} = -\frac{\partial E}{\partial\gamma}(\Psi(r,\cdot))(t).$$ However, I am not very firm in the theory of PDE, so I am not sure whether i can assume existence of $\Psi$, in particular if $\gamma$ is only piesewice differentiable. Also, from this approach it seems very unclear to me how a reparametrisation of $\gamma$ would influence the resulting geodesic and how we can keep $\gamma(0), \gamma(1)$ fixed.