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Degenerate Lagrange densities involving geometric objects

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Abstract

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of a symmetric affine connection, Γ hi j , and its first derivatives together with a symmetric tensor gi j, are investigated. In general, by variation of the Γ hi j , these equations will be of second order in Γ hi j . Necessary and sufficient conditions for these Euler-Lagrange equations to be of order one and zero in Γ hi j are obtained. It is shown that if the gi j may be regarded as independent then the only permissible zero order Euler-Lagrange equations are those which ensure that the Γ hi j are precisely the Christoffel symbols of the second kind.

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Communicated by R. A. Toupin

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Lovelock, D. Degenerate Lagrange densities involving geometric objects. Arch. Rational Mech. Anal. 36, 293–304 (1970). https://doi.org/10.1007/BF00249517

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  • DOI: https://doi.org/10.1007/BF00249517

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