On the Helmholtz operator of variational calculus in fibered-fibered manifolds

Abstract

A fibered-fibered manifold is a surjective fibered submersion π : Y → X between fibered manifolds. For natural numbers s ≥ r ≤ q an (r, s, q)th order Lagrangian on a fibered-fibered manifold π : Y → X is a base-preserving morphism λ : JY → ∧dimX T X. For p = max(q, s) there exists a canonical Euler morphism E(λ) : JY → VY ⊗ ∧dimX T X satisfying a decomposition property similar to the one in the fibered manifold case, and the critical fibered sections σ of Y are exactly the solutions of the Euler–Lagrange equation E(λ) ◦ jσ = 0. In the present paper, similarly to the fibered manifold case, for any morphism B : JY → VY ⊗ ∧m T X over Y , s ≥ r ≤ q, we define canonically a Helmholtz morphism H(B) : JY → VJY ⊗ VY ⊗ ∧dimX T X, and prove that a morphism B : JY → VY ⊗ ∧ T M over Y is locally variational (i.e. locally of the form B = E(λ) for some (r, s, p)th order Lagrangian λ) if and only if H(B) = 0, where p = max(s, q). Next, we study naturality of the Helmholtz morphism H(B) on fiberedfibered manifolds Y of dimension (m1,m2, n1, n2). We prove that any natural operator of the Helmholtz morphism type is cH(B), c ∈ R, if n2 ≥ 2. 0. Introduction. The first problem in variational calculus is to characterize critical values. It is known that the critical sections of a fibered manifold p : X → X0 with respect to an rth order Lagrangian λ : J X → dimX0T X0 can be characterized as the solutions of the so-called Euler– Lagrange equation. There exists a unique Euler map E(λ) : JX → V X⊗ dimX0T X0 over X satisfying some decomposition formula. Then the Euler–Lagrange equation is E(λ)◦jσ = 0 with unknown section σ (see [2]). The second problem is to characterize morphisms B : JX → V X ⊗ dimX0T X0 over X which are locally variational (i.e. locally of the form B = E(λ) for some rth order Lagrangian λ). In [3], for any natural number r and any morphism B : JY → V X ⊗ dimX0T X0 over 2000 Mathematics Subject Classification: Primary 58A20.