A first order prolongation of the conventional space. (English) Zbl 0867.58021

Matsyuk, R. Ya.

Janyška, Josef (ed.) et al., Differential geometry and applications. Proceedings of the 6th international conference, Brno, Czech Republic, August 28–September 1, 1995. Brno: Masaryk University. 403-415 (1996).

Let \(T^rM=J^r_0({\mathbb{R}},M)\) be the \(r\)-th order tangent bundle of a manifold \(M\). The author first constructs a map \(TP^2 M/G^2_m\to T^3M\), where \(P^2M(M,G^2_m)\) is the second order frame bundle of \(M\). A second order connection is defined as a map \(T^2 M\to TP^2M/G^2_m\) over the identity of \(TM\). Such a connection is called stable or quasi-stable, if it projects into the identity of \(T^2M\) 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].

Reviewer: I.Kolář (Brno)

MSC:

58E30	Variational principles in infinite-dimensional spaces
49N45	Inverse problems in optimal control

Keywords:

higher-order connection; inverse variational problem