## Of the structure of the Euler mapping.(English)Zbl 0337.33012

Let $$\mathbb R^n$$, $$\mathbb R^m$$ be real Euclidean spaces of dimension $$n, m$$, respectively, $$L(\mathbb R^n, \mathbb R^m)$$ the vector space of all linear mappings from $$\mathbb R^n$$ into $$\mathbb R^m$$, and $$L_s^2(\mathbb R^n, \mathbb R^n)$$ the vector space of all symmetric bilinear mappings from $$\mathbb R^n\times\mathbb R^n$$ into $$\mathbb R^n$$. For open subsets $$U\subset\mathbb R^n$$ and $$V\subset\mathbb R^n$$ define $$\mathcal J^1=U\times V\times L(\mathbb R^n, \mathbb R^m)$$ and $$\mathcal J^2 = U\times V\times L(\mathbb R^n, \mathbb R^n)\times L_s^2(\mathbb R^n, \mathbb R^n)$$. Assume $$\mathcal J^1$$ and $$\mathcal J^2$$ are differentiable manifolds in natural coordinates. Let $$L$$ be a Lagrange function and $$\mathcal E_\mu(L)=0$$, $$\mu=1,2,\ldots,m$$, the Euler equations defining the extremals associated with $$L$$. Certain sufficient conditions for the identical vanishing of the left side of the Euler equations are known. However, a complete description of the Lagrange functions satisfying the Euler 1-form $$\mathcal E_\mu(L)=0$$, $$(\mu=1)$$ is not known except when $$m=1$$. An attempt is made to provide such a description.
Let $$\mathcal L(\mathcal J^1)$$ and $$\Omega^1(\mathcal J^2)$$ denote the vector spaces of all Lagrange functions and Euler 1-forms on $$\mathcal J^2$$, respectively. An effort is made to study the kernel of the linear mapping $$\mathcal L(\mathcal J^1)\ni L\to \mathcal E(L)\in\Omega^1(\mathcal J^2)$$. After establishing several lemmas, the equivalence of $$\mathcal E(L)=0$$ to the existence of an $$n$$-form satisfying certain conditions is proved.
Reviewer: N. A. Warsi

### MSC:

 5.8e+16 Variational problems concerning extremal problems in several variables; Yang-Mills functionals 5.8e+31 Variational principles in infinite-dimensional spaces
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