Geodesic
On the projective Finsler metrizability and the integrability of Rapcsák equation
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 469-495.

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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 \hbox {2-acyclicity} of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.
DOI : 10.21136/CMJ.2017.0010-16
Classification : 49N45, 53C22, 53C60, 58E30
Keywords: Euler-Lagrange equation; metrizability; projective metrizability; geodesics; spray; formal integrability 
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     title = {On the projective {Finsler} metrizability and the integrability of {Rapcs\'ak} equation},
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Milkovszki, Tamás; Muzsnay, Zoltán. On the projective Finsler metrizability and the integrability of Rapcsák equation. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 469-495. doi : 10.21136/CMJ.2017.0010-16. http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0010-16/

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