Homogeneity and projective equivalence of differential equation fields

Mike Crampin; David Saunders

doi:10.3934/jgm.2012.4.27

Article Contents

2012, Volume 4, Issue 1: 27-47. Doi: 10.3934/jgm.2012.4.27

This issue Previous Article Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds Next Article Variational reduction of Lagrangian systems with general constraints

Homogeneity and projective equivalence of differential equation fields

Mike Crampin^1, and
David Saunders^2,

1.
Department of Mathematics, Ghent University, Krijgslaan 281, B-9000 Gent
2.
Department of Mathematics, Faculty of Science, The University of Ostrava, 30. dubna 22, 701 03 Ostrava

Received: September 2011

Revised: March 2012

Published: March 2012

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Abstract

We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are regular (in a natural sense that we define) form a projective equivalence class of homogeneous systems. We show further that the geodesics, or base integral curves, of projectively equivalent homogeneous differential equation fields are the same apart from orientation-preserving reparametrization; that is, homogeneous differential equation fields determine systems of paths.

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Mathematics Subject Classification: 34A26, 70H03, 70H50.

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References

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