Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 871the energy Lagrangian whose extremals are the harmonic maps between thesemi–Riemannian manifolds (R, h) and (M, g). At the same time, this Lagrangianis a basic object in the physical theory of bosonic strings (comparewith subsection 6.7 below).In above notations, taking U (1)(i) (t, x) as a d−tensor–field on J 1 (R, M)and F : R × M → R a smooth map, the more general Lagrangian functionL 2 : J 1 (R, M) → R defined byL 2 = h 11 (t)g ij (x)v i v j + U (1)(i) (t, x)vi + F (t, x) (5.200)is also a Kronecker h−regular Lagrangian. The relativistic rheonomic Lagrangianspace RL n = (J 1 (R, M), L 2 ) is called the autonomous relativisticrheonomic Lagrangian space of electrodynamics because, in the particularcase h 11 = 1, we recover the classical Lagrangian space of electrodynamics[Miron et. al. (1988); Miron and Anastasiei (1994)] which governs the movementlaw of a particle placed concomitantly into a gravitational field and anelectromagnetic one. From a physical point of view, the semi–Riemannianmetric h 11 (t) (resp. g ij (x)) represents the gravitational potentials of thespace R (resp. M), the d−tensor U (1)(i)(t, x) stands for the electromagneticpotentials and F is a function which is called potential function. The nondynamicalcharacter of spatial gravitational potentials g ij (x) motivates usto use the term of ‘autonomous’.More general, if we consider g ij (t, x) a d−tensor–field on J 1 (R, M),symmetric, of rank n and having a constant signature on J 1 (R, M), we candefine the Kronecker h−regular Lagrangian function L 3 : J 1 (R, M) → R,settingL 3 = h 11 (t)g ij (t, x)v i v j + U (1)(i) (t, x)vi + F (t, x). (5.201)The pair RL n = (J 1 (R, M), L 3 ) is a relativistic rheonomic Lagrangian spacewhich is called the non–autonomous relativistic rheonomic Lagrangianspace of electrodynamics. Physically, we remark that the gravitational potentialsg ij (t, x) of the spatial manifold M are dependent of the temporalcoordinate t, emphasizing their dynamical character.5.7.2 Canonical Nonlinear ConnectionsLet us consider h = (h 11 ) a fixed semi–Riemannian metric on R and arheonomic Lagrangian space RL n = (J 1 (R, M), L), where L is a Kroneckerh−regular Lagrangian function. Let [a, b] ⊂ R be a compact interval in