Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 877Let Γ = (M (i)(1)1 N (i)(1)j ) be a nonlinear connection on J 1 (R, M). Let usconsider the geometrical objects,δδt = ∂ (j) ∂−M∂t(1)1∂v j ,δδx i = ∂ (j) ∂−N∂xi (1)i∂v j ,δvi = dy i +M (i) (i)(1)1dt+N(1)j dxj .One can deduce that the set of vector–fields { δδt , δδx i ,∂∂v i }⊂ X (J 1 (R, M))and of covector–fields {dt, dx i , δv i } ⊂ X ∗ (J 1 (R, M)) are dual bases. Theseare called the adapted bases on J 1 (R, M), determined by the nonlinearconnection Γ. The big advantage of the adapted bases is that the transformationlaws of its elements are simple and natural. The transformation lawsof the elements of the adapted bases attached to the nonlinear connectionΓ areδδt = d¯t δdt δ¯t ,dt = dtd¯t d¯t,δδx i = ∂¯xj δ∂x i δ¯x j ,dx i = ∂xi∂¯x j d¯xj ,5.8 Jets and Action Principles∂∂v i = ∂¯xj dt δ∂x i d¯t δ¯v j ,δv i = ∂xi d¯t∂¯x j dt δ¯vj .Recall that in the classical calculus of variations one studies functionals ofthe form∫F L (z) = L(x, z, ∇z) dx, (with Ω ⊂ R n ), (5.211)Ωwhere x = (x 1 , . . . , x n ), dx = dx 1 ∧ · · · ∧ dx n , z = z(x) ∈ C 1 (¯Ω), and theLagrangian L = L(x, z, p) is a smooth function of x, z, and p = (p 1 , . . . , p n ).The corresponding Euler–Lagrangian equation, describing functions z(x)that are stationary for such a functional, is represented by the second–order PDE [Bryant et al. (2003)]∆z(x) = F ′ (z(x)).For example, we may identify a function z(x) with its graph N ⊂ R n+1 ,and take the LagrangianL = √ 1 + ||p|| 2 ,whose associated functional F L (z) equals the area of the graph, regardedas a hypersurface in Euclidean space. The Euler–Lagrangian equation describingfunctions z(x) stationary for this functional is H = 0, where H isthe mean curvature of the graph N.