classical mechanics - Are the Hamiltonian and Lagrangian always convex functions?

The Hamiltonian and Lagrangian are related by a Legendre transform: $$ H(\mathbf{q}, \mathbf{p}, t) = \sum_i \dot q_i p_i - \mathcal{L}(\mathbf{q}, \mathbf{\dot q}, t). $$ For this to be a Legendre transform, $H$ must be convex in each $p_i$ and $\mathcal{L}$ must be convex in each $\dot q_i$.

Of course this is the case for simple examples such as a particle in a potential well, or a relativistic particle moving inertially. However, it isn't obvious to me that it will always be the case for an arbitrary multi-component system using some complicated set of generalised coordinates.

Is this always the case? If so, is there some physical argument from which it can be shown? Or alternatively, are there cases where these convexity constraints don't hold, and if so what happens then?