This paper deals with integrability issues of the Euler-Lagrange equations associated to a variational problem, where the energy function depends on acceleration and drag. Although the motivation came from applications to path planning of underwater robot manipulators, the approach is rather theoretical and the main difficulties result from the fact that the power needed to push an object through a fluid increases as the cube of its speed.
Let $\mu \colon FX \to X$ be a principal bundle of frames with the structure group ${\rm Gl}_{n}(\mathbb R)$. It is shown that the variational problem, defined by ${\rm Gl}_{n}(\mathbb R)$-invariant Lagrangian on $J^{r} FX$, can be equivalently studied on the associated space of connections with some compatibility condition, which gives us order reduction of the corresponding Euler-Lagrange equations.