Ivancevic_Applied-Diff-Geom

882 Applied Differential Geometry: A Modern Introductionalso admits certain functionals depending on second derivatives of z(x),because there may be dp i −terms in Λ. Later, we will restrict attention to aclass of functionals which, possibly after a contact transformation, can beexpressed without second derivatives.Suppose given a Lagrangian Λ ∈ Ω n (M) on a contact manifold (M, I),and a fixed–boundary variation of Legendre submanifold F : N × [0, 1] →M; we wish to calculate d dt (∫ N tΛ).To do this, first recall the calculation of the Poincaré–Cartan form forthe equivalence class [Λ] ∈ ¯H n . Because I n+1 = Ω n+1 (M), we can write[Bryant et al. (2003)]and thendΛ = θ ∧ α + dθ ∧ β = θ ∧ (α + dβ) + d(θ ∧ β),Π = θ ∧ (α + dβ) = d(Λ − θ ∧ β). (5.213)We are looking for conditions on a Legendre submanifold f : N ↩→ M tobe stationary for [Λ] under all fixed–boundary variations, in the sense thatd ∣dt t=0( ∫ N tΛ) = 0 whenever F | t=0 = f. We calculate∫ ∫∫∫∂ t Λ = ∂ t (Λ − θ ∧ β) = L ∂t (Λ − θ ∧ β) = ∂ t ⌋Π.N t N t N t N tOne might express this result asδ(F Λ ) N (v) = int N v⌋f ∗ Π,where the variational vector–field v, lying in the space Γ 0 (f ∗ T M) of sectionsof f ∗ T M vanishing along ∂N, plays the role of ∂ t . The condition Π ≡0(mod{I}) allows us to write Π = θ ∧ Ψ for some n−form Ψ, not uniquelydetermined, and we have [Bryant et al. (2003)]∫ddt∣ Λ = g ft=0∫N ∗ Ψ,twhere g = (∂ t ⌋F ∗ θ)| t=0 . It was shown previously that this g could locallybe chosen arbitrarily in the interior N o , so the necessary and sufficientcondition for a Legendre submanifold f : N ↩→ M to be stationary for F Λis that f ∗ Ψ = 0.In the particular classical situation where M = {(x i , z, p i )}, θ = dz −p i dx i , and Λ = L(x, z, p)dx, we havedΛ = L z θ ∧ dx + L pi dp i ∧ dx = θ ∧ L z dx − dθ ∧ L pi dx (i) ,N