From Euler-Lagrange equations to canonical nonlinear connections

M. Neagu

Str. Lamaitei, Nr. 66, Bl. 93, Sc. G, Ap. 10,  Brasov, BV 500371, Romania


The aim of this paper is to construct a canonical nonlinear connection $\Gamma =(M_{(\alpha )\beta }^{(i)},    N_{(\alpha )j}^{(i)})$ on the 1-jet space $J^{1}(T,M)$ from the Euler-Lagrange equations of the quadratic multi-time Lagrangian function

$$L=h^{\alpha \beta }(t)g_{ij}(t,x)x_{\alpha }^{i}x_{\beta}^{j}+U_{(i)}^{(\alpha )}(t,x)x_{\alpha }^{i}+F(t,x)\,.$$

AMSclassification. 53B40, 53C60, 53C80.

Keywords.  1-jet fibre bundles, nonlinear connections, quadratic Lagrangian functions.