Ivancevic_Applied-Diff-Geom

872 Applied Differential Geometry: A Modern Introductionthe temporal manifold R. In this context, we can define the energy actionfunctional of RL n , settingE : C ∞ (R, M) → R, E(c) =∫ baL(t, x i , v i ) √ |h|dt,where the smooth curve c is locally expressed by (t) → (x i (t)) and v i = dxidt .The extremals of the energy functional E verifies the Euler–Lagrangianequations2G (1)(1)(i)(j) ẍj +∂2 L∂x j ∂v i ẋj − ∂L∂x i + ∂2 L∂t∂v i + ∂L∂v i H1 11 = 0,(i = 1, ..., n),(5.202)where H 1 11 are the Christoffel symbols of the semi–Riemannian metric h 11 .Taking into account the Kronecker h−regularity of the Lagrangian functionL, it is possible to rearrange the Euler–Lagrangian equations (5.202)of the Lagrangian L = L √ |h|, in the Poisson form [Neagu and Udrişte(2000a)]∆ h x i + 2G i (t, x i , v i ) = 0, (i = 1, ..., n), where (5.203)∆ h x i = h 11 { ẍ i − H11v 1 i} , v i = ẋ i ,{2G i = gii ∂ 2 L2 ∂x j ∂v i vj − ∂L∂x i + ∂2 L∂t∂v i + ∂L}∂v i H1 11 + 2g ij h 11 H11v 1 j .Denoting G (r)(1)1 = h 11G r , the geometrical object G = (G (r)(1)1) is a spatialspray on J 1 (R, M). By a direct calculation, we deduce that the localgeometrical entities of J 1 (R, M)2S k = gki22H k = gki2{ ∂ 2 L∂x j ∂v i vj − ∂L }∂x i ,{ ∂ 2 L∂t∂v i + ∂L∂v i H1 11verify the following transformation rules2S p r ∂xp ∂xp d¯t ∂¯x= 2 ¯S l γ+ h11∂¯xr∂¯x l dt ∂x j vj ,2J p = 2J ¯r∂xp ∂xp d¯t ∂¯v l− h11∂¯xr∂¯x l dt ∂t .}, 2J k = h 11 H 1 11v j ,2H p = 2r ∂xp ∂xp¯H + h11∂¯xrd¯t∂¯x l dt∂¯v l∂t ,Consequently, the local entities 2G p = 2S p + 2H p + 2J p can be modified