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.

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

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 2010): 49N45, 58E30, 53C60, 53C22

DOI: 10.21136/CMJ.2017.0010-16

Full text available as PDF.


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