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The inverse problem of the calculus of variations for Finsler structures. (English) Zbl 0585.53019
Let X be a differentiable manifold and TX the total space of its tangent bundle. A connection $$\Gamma$$ on TX of coefficients $$\Gamma^ i_{jk}(x,\dot x)$$ is called variational if a) there exists $$\gamma^ i_{jk}(x,\dot x)$$ such that $$\gamma^ i_{jk}\dot x^ j\dot x^ k=\Gamma^ j_{jk}\dot x^ j\dot x^ k;$$ b) there exists a regular mapping g: TX$$\to T^ 0_ 2X$$ over $$id_ X$$ with components $$g_{ij}(x,\dot x)$$ such that $\epsilon_ i=-g_{im}(\ddot x^ m+\gamma^ m_{pq}\dot x^ p\dot x^ q)$ is the Euler-Lagrange covector of a Lagrangian $$L(x,\dot x,\ddot x).$$ $$\Gamma$$ is called metrizable if g is regular, symmetric, positive-definite and satisfies: $$(\partial g_{ij}/\partial \dot x^ k)\dot x^ j=0$$, $$\nabla_{\Gamma}g=0$$. The important results of this paper are given by: 1) $$\Gamma$$ is variational if and only if it is metrizable; 2) each metrizable connection $$\Gamma$$ on TX is a Cartan connection of a Finsler space $$F^ n$$ on X.
Reviewer: R.Miron

##### MSC:
 53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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