Geodesic
On the projective Finsler metrizability and the integrability of Rapcsák equation
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 469-495 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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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

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