We are given an interval $I \subset \mathbb{R},$ the space of $q-$absolutely continuous functions $AC^q(I,\mathbb{R}^{n})$ which satisfy continuity of the function and the existence of its $q-$integrable weak derivative.
Let $F: AC^q(I,\mathbb{R}^{n}) \rightarrow \mathbb{R}, F(\gamma) = \int_{t_0}^{t_1} L(\gamma(t), \dot{\gamma}(t))dt$ be a functional on $AC^q(I, \mathbb{R}^{n})$ with $\gamma \in AC^q(I, \mathbb{R}^{n}), t_0, t_1 \in I.$
I saw in different sources that convexity of the Lagrangian $L: \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ in the argument of the derivative $\dot{\gamma}(t)$ ensures the minimizing curve $\gamma$ does not "oscillate itself to death" in the attempt to minimize the functional.
Why does convexity prevent oscillations? Any intuition and complete proof or reference to one will be appreciated.