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Let $C$ be the set of smooth curves from the unit interval into $\mathbb{R}^n$. Let $f : C \rightarrow \mathbb{R}$ be a functional on these curves given by $f(x) := \int_0^1 L(x,\dot{x}) dt$. Define the Hessian of $f$ at $x(t)$ as $f_{**}(V,W)$ which a bi-linear form on vector fields along $x(t)$. This is defined by first taking a variation of curves $x(h_1, h_2)$ satisfying $x(0,0) = x$ and $\frac{\partial x(h_1, h_2)}{\partial h_1}|_{(0,0)} = V$ and $\frac{\partial x(h_1, h_2)}{\partial h_2}|_{(0,0)} = W$. Then we set $f_{**}(V,W) = \frac{\partial^2 f(x(h_1, h_2))}{\partial h_1 \partial h_2}|_{(0,0)}$

What conditions would I need on $f$ to show the Hessian of $f$ is positive or negative definite. ie) Are there any standard techniques to show $f$ is either a minimum or a maximum at $x$. I've seen a theorem for geodesics which basically says that if there are no conjugate points along $x(t)$, then we are guaranteed the geodesic will be a local minimum but the proof for this usually involves geometrical argument specifically for this functional. Does such a theorem generalise for the equivalent notion of conjugacy for $f$ given we place suitable constraints on $f$?

Anything to enlighten me on this topic would be appreciated. Thanks.

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