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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

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
Full Text: EuDML
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