Let $T^{r}M=J_0^r(ℝ,M)$ be the $r$-th order tangent bundle of a manifold $M$. The author first constructs a map $T{P}^{2}M/G_m^2\to T^{3}M$, where $P^{2}M(M,G_m^2)$ is the second order frame bundle of $M$. A second order connection is defined as a map $T^{2}M\to T{P}^{2}M/G_m^2$ over the identity of $TM$. Such a connection is called stable or quasi-stable, if it projects into the identity of $T^{2}M$ or into the space of contact (1,2)-elements, respectively. Using these geometric ideas, the author solves a special type of the inverse variational problem in pseudo-Euclidean 3-space.

For the entire collection see [Zbl 0847.00040].