Ivancevic_Applied-Diff-Geom

870 Applied Differential Geometry: A Modern Introduction5.7.1 Relativistic Rheonomic Lagrangian SpacesIn order to develop the time–dependent Lagrangian geometry, following[Neagu and Udrişte (2000a); Udriste (2000); Neagu (2002); Neagu andUdrişte (2000b); Neagu (2000)], we consider L : J 1 (R, M) → R to be asmooth Lagrangian function on the 1–jet bundle J 1 (R, M) → R, locallyexpressed by (t, x i , v i ) ↦→ L(t, x i , v i ). The so–called vertical fundamentalmetrical d−tensor of L is defined byR.(i)(j) = 1 ∂ 2 L2 ∂v i ∂v j . (5.197)G (1)(1)Let h = (h 11 ) be a semi–Riemannian metric on the temporal manifoldA Lagrangian function L : J 1 (R, M) → R whose vertical fundamentalmetrical d−tensor is of the formG (1)(1)(i)(j) (t, xk , v k ) = h 11 (t)g ij (t, x k , v k ), (5.198)where g ij (t, x k , v k ) is a d−tensor on J 1 (R, M), symmetric, of rank n andhaving a constant signature on J 1 (R, M), is called a Kronecker h−regularLagrangian function, with respect to the temporal semi–Riemannian metrich = (h 11 ).A pair RL n = (J 1 (R, M), L), where n = dim M, which consists ofthe 1−jet space J 1 (R, M) and a Kronecker h−regular Lagrangian functionL : J 1 (T, M) → R is called a relativistic rheonomic Lagrangian space.In our geometrization of the time–dependent Lagrangian function L thatwe will construct, all entities with geometrical or physical meaning willbe directly arisen from the vertical fundamental metrical d−tensor G (1)(1)(i)(j) .This fact points out the metrical character (see [Gotay et. al. (1998)])and the naturalness of the subsequent relativistic rheonomic Lagrangiangeometry.For example, suppose that the spatial manifold M is also equippedwith a semi–Riemannian metric g = (g ij (x)). Then, the time–dependentLagrangian function L 1 : J 1 (R, M) → R defined byL 1 = h 11 (t)g ij (x)v i v j (5.199)is a Kronecker h−regular time–dependent Lagrangian function. Consequently,the pair RL n = (J 1 (R, M), L 1 ) is a relativistic√rheonomic Lagrangianspace. We underline that the Lagrangian L 1 = L 1 |h11 | is exactly