Canonical Nonlinear Connections on Jet Bundles of First Order

The aim of this paper is to open the problem of construction of a nonlinear connection $\Gamma=(M^{(i)}_{(\alpha)\beta}, N^{(i)}_{(\alpha)j})$ on the jet bundle of first order $J^1(T,M)$, which to be canonically produced by a Kronecker product vertical metrical d-tensor $G^{(\alpha)(\beta)}_{(i)(j)}=h^{\alpha\beta}g_{ij}$, possibly provided by multi-time dependent quadratic Lagrangians coming from various branches of theoretical physics: bosonic strings theory, electrodynamics or elasticity.