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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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##### References:
 [1] KRUPKA D.: Local invariants of a linear connection. Colloquia Math. Soc. J. Bolyai, 31. Differential Geometry, Budapest 1979, North Holland, 1982, 349-369. [2] KRUPKA D.: On the local structure of the Euler-Lagrange mapping of the calculus of variations. Proc. Conf. on Diff. Geom. and Appl., Nové Město na Moravě, September 1980; Charles Univ. of Prague, 1982, 181-188. [3] ЛАПТЄВ Б. Л.: Производная Ли для обьєктов, являющихся функциєй направлєния, Изв. физ.-мат. об-ва при Казанск. унив. 3, 10, 1938, 3-38. [4] RUND H.: The Differential Geometry of Finsler Spaces. Springer, Berlin 1959. · Zbl 0087.36604 [5] САТТАРОВ А. Э.: Экстрємали мєтричєского просгранства опорных вєкторных плотностєй. Изв. вузов Матєматика 9, 1977, 119-121. [6] TONTT E.: Variational formulation of nonlinear differential equations, I., II. Bull. Classe Sciences Acad. R. de Belgique 55, 1969, 137-165; 262-278.
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