  Exploiting our previous results on higher order controlled Lagrangians in
[Nonlinear Anal. {\bf 207} (2021), 112263], we derive here an analogue of the
classical first order Pontryagin Maximum Principle (PMP) for cost minimising
problems subjected to higher order differential constraints $\frac{d^k
x^j}{dt^k} = f^j\big(t, x(t), \frac{d x}{dt}(t), \ldots, \frac{d^{k-1}
x}{dt^{k-1}}(t), u(t)\big)$, $t \in [0,T]$, where $u(t)$ is a control curve in
a compact set $K \subset \mathbb R^m$. This result and its proof can be
considered as a detailed illustration of one of the claims of that previous
paper, namely that the results of that paper, originally established in a
smooth differential geometric framework, yield directly properties holding
under much weaker and more common assumptions. In addition, for further
clarifying our motivations, in the last section we display a couple of quick
indications on how the two-step approach of this paper (i.e., a preliminary
easy-to-get differential geometric discussion followed by a refining analysis
to weaken the regularity assumptions) might be fruitfully exploited also in the
context of control problems governed by partial differential equations or in
studies on the dynamics of controlled mechanical systems.
