This question arose out of studying "nice" conditions on Lagrangians. Let $L : \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$ (let us write $L(x, v, t)$) be convex in $v$ (for any $x, t$). Is it true that any extrema of $L$ must be either global minima, or a saddle point? In particular, $L$ cannot achieve a maximum?
I ask this because the Euler-Lagrange equations are equivalent to being a critical point of the action functional, but generally we want to minimize (at least, as far as I know) the action, and so I'm wondering under what conditions we get critical point $\implies$ minimum. I've seen convexity in the $v$-variable as an assumption before. Why in this variable? What about $x$, or $t$?
Many thanks for any help!
The answer to your question is negative. If $L(x,v,t)$ has a maximum at $(x^*,v^*,t^*)$, then the convex function $f(t) = (x^*,v^*,t)$ has a maximum at $t^*$, which a convex function with domain $\mathbb{R}$ cannot have (unless the function is constant).