The Geometry of MultiLagrange Spaces
Mircea Neagu, Constantin Udrişte · arXiv (Cornell University) · 2000
Section 1 presents the actual mathematical context in which is placed the geometry of multiLagrange spaces developped in this paper. This is a new geometry of the 1-jet space J 1 (T, M) naturally attached to a Kronecker h-regular Lagrangian with partial derivatives of order one. In Section 2 is introduced the notion of multiLagrange space ML n p = (J 1 (T, M), L) and is proved the theorem of characterization of these spaces. In Section 3 is constructed the canonical nonlinear connection Γ = (M (i), N(i) (α)β (α)j) on J1 (T, M), naturally induced by the Lagrangian L of the multiLagrangian space ML n p. At the same time, in Section 3 is offered a geometrical interpretation of the extremals of L. Section 4 is dedicated to the study of the important Cartan canonical connection CΓ of a multiLagrange space ML n p which allows to construct the subsequent multiLagrangian theory of physical fields [7]. The main torsion and curvature d-tensors of the Cartan canonical connection CΓ are also described in Section 4.