Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 469-495, 2017


On the projective Finsler metrizability and the integrability of Rapcsák equation

Tamás Milkovszki, Zoltán Muzsnay

Received January 10, 2016.  First published May 12, 2017.

Abstract:  A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.
Keywords:  Euler-Lagrange equation; metrizability; projective metrizability; geodesics; spray; formal integrability
Classification MSC:  49N45, 58E30, 53C60, 53C22
DOI:  10.21136/CMJ.2017.0010-16


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Affiliations:   Tamás Milkovszki, Zoltán Muzsnay, Institute of Mathematics, University of Debrecen, Egyetem tér 1, H-4032 Debrecen, Hungary, e-mail: milkovszki@science.unideb.hu, muzsnay@science.unideb.hu

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