On the projective Finsler metrizability and the integrability of Rapcsák equation

Czechoslovak Mathematical Journal

Tamás Milkovszki, Zoltán Muzsnay¹ 

关键词: Euler-Lagrange equation;    metrizability;    projective metrizability;    geodesics;    spray;    formal integrability;   

DOI  :  10.21136/CMJ.2017.0010-16

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

  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.

【 授权许可】

Unknown