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

W. M. Mikulski. "On the Helmholtz operator of variational calculus in fibered-fibered manifolds." Annales Polonici Mathematici 90.1 (2007): 59-76. <http://eudml.org/doc/280515>.

@article{W2007,
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 $λ: J^\{r,s,q\}Y → ⋀^\{dim X\}T*X$. For p= max(q,s) there exists a canonical Euler morphism $(λ): J^\{r+s,2s,r+p\}Y → *Y⊗ ⋀^\{dim X\}T*X$ 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 $(λ) ∘ j^\{r+s,2s,r+p\}σ = 0$. In the present paper, similarly to the fibered manifold case, for any morphism $B:J^\{r,s,q\}Y → *Y ⊗ ⋀^\{m\}T*X$ over Y, s ≥ r ≤ q, we define canonically a Helmholtz morphism $(B): J^\{s+p,s+p,2p\}Y → *J^\{r,s,r\}Y ⊗ *Y ⊗ ⋀^\{dim X\}T*X$, and prove that a morphism $B:J^\{r+s,2s,r+p\} Y → *Y ⊗ ⋀ T*M$ over Y is locally variational (i.e. locally of the form B = (λ) for some (r,s,p)th order Lagrangian λ) if and only if (B) = 0, where p = max(s,q). Next, we study naturality of the Helmholtz morphism (B) on fibered-fibered manifolds Y of dimension (m₁,m₂,n₁,n₂). We prove that any natural operator of the Helmholtz morphism type is c(B), c ∈ ℝ, if n₂≥ 2.},
author = {W. M. Mikulski},
journal = {Annales Polonici Mathematici},
keywords = {fibered-fibered manifold; -jet prolongation bundle; order Lagrangian; Euler morphism; Helmholtz morphism; natural operator},
language = {eng},
number = {1},
pages = {59-76},
title = {On the Helmholtz operator of variational calculus in fibered-fibered manifolds},
url = {http://eudml.org/doc/280515},
volume = {90},
year = {2007},
}

TY - JOUR
AU - W. M. Mikulski
TI - On the Helmholtz operator of variational calculus in fibered-fibered manifolds
JO - Annales Polonici Mathematici
PY - 2007
VL - 90
IS - 1
SP - 59
EP - 76
AB - 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 $λ: J^{r,s,q}Y → ⋀^{dim X}T*X$. For p= max(q,s) there exists a canonical Euler morphism $(λ): J^{r+s,2s,r+p}Y → *Y⊗ ⋀^{dim X}T*X$ 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 $(λ) ∘ j^{r+s,2s,r+p}σ = 0$. In the present paper, similarly to the fibered manifold case, for any morphism $B:J^{r,s,q}Y → *Y ⊗ ⋀^{m}T*X$ over Y, s ≥ r ≤ q, we define canonically a Helmholtz morphism $(B): J^{s+p,s+p,2p}Y → *J^{r,s,r}Y ⊗ *Y ⊗ ⋀^{dim X}T*X$, and prove that a morphism $B:J^{r+s,2s,r+p} Y → *Y ⊗ ⋀ T*M$ over Y is locally variational (i.e. locally of the form B = (λ) for some (r,s,p)th order Lagrangian λ) if and only if (B) = 0, where p = max(s,q). Next, we study naturality of the Helmholtz morphism (B) on fibered-fibered manifolds Y of dimension (m₁,m₂,n₁,n₂). We prove that any natural operator of the Helmholtz morphism type is c(B), c ∈ ℝ, if n₂≥ 2.
LA - eng
KW - fibered-fibered manifold; -jet prolongation bundle; order Lagrangian; Euler morphism; Helmholtz morphism; natural operator
UR - http://eudml.org/doc/280515
ER -