Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 879geometrical approach, we note the fact that Lagrangian is invariant underthe large class of contact transformations. Also, note that the LagrangianL determines the functional F L , but not vice versa. To see this, observethat if we add to L(x, z, p) a divergence term and considerL ′ (x, z, p) = L(x, z, p) + ∑ (∂ x iK i (x, z) + ∂ z K i (x, z)p i )for functions K i (x, z), then by the Green’s Theorem, the functionals F Land F L ′ differ by a constant depending only on values of z on ∂Ω. L andL ′ have the same Euler–Lagrangian equations.Also, there is a relationship between symmetries of a Lagrangian Land conservation laws for the corresponding Euler–Lagrangian equations,described by the Noether Theorem. A subtlety here is that the group ofsymmetries of an equivalence class of Lagrangians may be strictly largerthan the group of symmetries of any particular representative. We willinvestigate how this discrepancy is reflected in the space of conservationlaws, in a manner that involves global topological issues.Finally, one considers the second variation δ 2 F L , analogous to the Hessianof a smooth function, usually with the goal of identifying local minimaof the functional. There has been a great deal of analytic work done in thisarea for classical variational problems, reducing the problem of local minimizationto understanding the behavior of certain Jacobi operators, butthe geometrical theory is not as well–developed as that of the first variationand the Euler–Lagrangian equations.Now we turn to multi–index notation [Griffiths (1983); Bryant et al.(2003); Choquet-Bruhat and DeWitt-Morete (1982)]. An exterior differentialsystem (EDS) is a pair (M, E) consisting of a smooth manifold Mand a homogeneous, differentially closed ideal E ⊆ Ω ∗ (M) in the algebraof smooth differential forms on M. Some of the EDSs that we study aredifferentially generated by the sections of a smooth subbundle I ⊆ T ∗ Mof the cotangent bundle of M; this subbundle, and sometimes its space ofsections, is called a Pfaffian system on M. It will be useful to use the notation{α, β, . . .} for the (two–sided) algebraic ideal generated by forms α,β,. . . , and to use the notation {I} for the algebraic ideal generated by thesections of a Pfaffian system I ⊆ T ∗ M. An integral manifold of an EDSdef(M, E) is a submanifold immersion ι : N ↩→ M for which ϕ N = ι ∗ ϕ = 0for all ϕ ∈ E. Integral manifolds of Pfaffian systems are defined similarly.A differential form ϕ on the total space of a fibre bundle π : E → Bis said to be semibasic if its contraction with any vector–field tangent to