  We investigate the optimal mass transport problem associated to the following
"ballistic" cost functional on phase space $M\times M^*$, $$ b_T(v,
x):=\inf\{\langle v, \gamma (0)\rangle +\int_0^TL(\gamma (t), {\dot
\gamma}(t))\, dt, \gamma \in C^1([0, T), M), \gamma(T)=x\}, $$ where
$M=\mathbb{R}^d$, $T>0$, and $L:M\times M \to \mathbb{R}$ is a Lagrangian that
is jointly convex in both variables. Under suitable conditions on the initial
and final probability measures, we use convex duality \`a la Bolza and
Monge-Kantorovich theory to lift classical Hopf-Lax formulae from state space
to Wasserstein space. This allows us to relate optimal transport maps for the
ballistic cost to those associated with the fixed-end cost defined on $M\times
M$ by $$ c_T(x,y):=\inf\{\int_0^TL(\gamma(t), {\dot \gamma}(t))\, dt, \gamma\in
C^1([0, T), M), \gamma(0)=x, \gamma(T)=y\}. $$ We also point to links with the
theory of mean field games.
