Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 881Let (M, I) be a contact manifold of dimension 2n + 1, and assume thatI is generated by a global, non–vanishing section θ ∈ Γ(I); this assumptiononly simplifies our notation, and would in any case hold on a double–coverof M. Sections of I generate the contact differential idealI = {θ, dθ} ⊂ Ω ∗ (M)in the exterior algebra of differential forms on M. A Legendre submanifoldof M is an immersion ι : N ↩→ M of an nD submanifold N such thatι ∗ θ = 0 for any contact form θ ∈ Γ(I); in this case ι ∗ dθ = 0 as well,so a Legendre submanifold is the same thing as an integral manifold ofthe differential ideal I. In Pfaff coordinates with θ = dz − p i dx i , onesuch integral manifold is N 0 = {z = p i = 0}. To see other Legendresubmanifolds ‘near’ this one, note than any submanifold C 1 −close to N 0satisfies the independence condition [Bryant et al. (2003)]dx 1 ∧ · · · ∧ dx n ≠ 0,and can therefore be described locally as a graphIn this case, we haveN = {(x i , z(x), p i (x))}.θ| N = 0 iff p i (x) = ∂ x iz(x).Therefore, N is determined by the function z(x), and conversely, everyfunction z(x) determines such an N; we informally say that ‘the genericLegendre submanifold depends locally on one arbitrary function of n variables’.Legendre submanifolds of this form, with dx| N ≠ 0, are calledtransverse.Now, we are interested in functionals given by triples (M, I, Λ), where(M, I) is a (2n + 1)D contact manifold, and Λ ∈ Ω n (M) is a differentialform of degree n on M; such a Λ will be referred to as a Lagrangian on(M, I) [Bryant et al. (2003)]. We then define a functional on the set ofsmooth, compact Legendre submanifolds N ⊂ M, possibly with boundary∂N, by∫F Λ (N) = Λ.The classical variational problems described above may be recovered fromthis notion by taking M = J 1 (R n , R) ∼ = R 2n+1 with coordinates (x i , z, p i ),I generated by θ = dz − p i dx i , and Λ = L(x i , z, p i )dx. This formulationN