Ivancevic_Applied-Diff-Geom

878 Applied Differential Geometry: A Modern IntroductionTo study these Lagrangians and Euler–Lagrangian equations geometrically,we have to choose a class of admissible coordinate changes, andthere are four natural candidates. In increasing order of generality, theyare [Bryant et al. (2003)]:• Classical transformations, of the form x ′ = x ′ (x), z ′ = z ′ (z); in thissituation, we think of (x, z, p) as coordinates on the space J 1 (R n , R) of1−jets of maps R n → R.• Gauge transformations, of the form x ′ = x ′ (x), z ′ = z ′ (x, z); here, wethink of (x, z, p) as coordinates on the space of 1−jets of sections ofa bundle R n+1 → R n , where x = (x 1 , . . . , x n ) are coordinates on thebase R n and z ∈ R is a fibre coordinate.• Point transformations, of the form x ′ = x ′ (x, z), z ′ = z ′ (x, z); here, wethink of (x, z, p) as coordinates on the space of tangent hyperplanes{dz − p i dx i } ⊥ ⊂ T (x i ,z)(R n+1 )of the manifold R n+1 with coordinates (x 1 , . . . , x n , z).• Contact transformations, of the form x ′ = x ′ (x, z, p), z ′ = z ′ (x, z, p),p ′ = p ′ (x, z, p), satisfying the equation of differential 1−formsfor some function f(x, z, p) ≠ 0.dz ′ − p ′ idx i′ = f · (dz − p i dx i )Classical calculus of variations primarily concerns the following featuresof the functional F L (5.211).The first variation δF L (z) is analogous to the derivative of a function,where z = z(x) is thought of as an independent variable in an infinite–dimensional space of functions. The analog of the condition that a pointbe critical is the condition that z(x) be stationary for all fixed–boundaryvariations. Formally, we writeδF L (z) = 0,which will give us a second–order scalar PDE for the unknown functionz(x) of the form∂ z L − ∂ x i(∂ pi L) = 0, (5.212)namely the Euler–Lagrangian equation of the Lagrangian L(x, z, p).In this section we will study the PDE (5.212) in an invariant, geometricalsetting, following [Bryant et al. (2003)]. As a motivation for this