# American Institute of Mathematical Sciences

March  2012, 4(1): 27-47. doi: 10.3934/jgm.2012.4.27

## Homogeneity and projective equivalence of differential equation fields

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

Received  September 2011 Revised  March 2012 Published  April 2012

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.
Citation: Mike Crampin, David Saunders. Homogeneity and projective equivalence of differential equation fields. Journal of Geometric Mechanics, 2012, 4 (1) : 27-47. doi: 10.3934/jgm.2012.4.27
##### References:
 [1] I. Bucataru, O. A. Constantinescu and M. F. Dahl, A geometric setting for systems of ordinary differential equations,, Int. J. Geom. Methods Mod. Phys., 8 (2011), 1291.  doi: 10.1142/S0219887811005701.  Google Scholar [2] M. Crampin, Homogeneous systems of higher-order ordinary differential equations,, Communications in Mathematics, 18 (2010), 37.   Google Scholar [3] M. Crampin and D. J. Saunders, The Hilbert-Carathéodory and Poincaré-Cartan forms for higher-order multiple-integral variational problems,, Houston J. Math., 30 (2004), 657.   Google Scholar [4] F. Faà di Bruno, Sullo sviluppo delle Funzioni,, Annali di Scienze Matematiche e Fisiche, 6 (1855), 479.   Google Scholar [5] I. Kolář, P. W. Michor and J. Slovak, "Natural Operations in Differential Geometry,", Springer-Verlag, (1993).   Google Scholar [6] B. S. Kruglikov and V. V. Lychagin, Geometry of differential equations,, in, (2008), 725.   Google Scholar [7] R. Ya. Matsyuk, Higher order variational origin of the Dixon's system and its relation to the quasi-classical 'Zitterbewegung' in general relativity,, preprint, ().   Google Scholar [8] J. Muñoz Masqué and I. M. Pozo Coronado, Parameter-invariant second-order variational problems in one varaiable,, J. Phys. A, 31 (1998), 6225.  doi: 10.1088/0305-4470/31/29/014.  Google Scholar [9] J. J. Stoker, "Differential Geometry,", Pure and Applied Mathematics, (1969).   Google Scholar

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##### References:
 [1] I. Bucataru, O. A. Constantinescu and M. F. Dahl, A geometric setting for systems of ordinary differential equations,, Int. J. Geom. Methods Mod. Phys., 8 (2011), 1291.  doi: 10.1142/S0219887811005701.  Google Scholar [2] M. Crampin, Homogeneous systems of higher-order ordinary differential equations,, Communications in Mathematics, 18 (2010), 37.   Google Scholar [3] M. Crampin and D. J. Saunders, The Hilbert-Carathéodory and Poincaré-Cartan forms for higher-order multiple-integral variational problems,, Houston J. Math., 30 (2004), 657.   Google Scholar [4] F. Faà di Bruno, Sullo sviluppo delle Funzioni,, Annali di Scienze Matematiche e Fisiche, 6 (1855), 479.   Google Scholar [5] I. Kolář, P. W. Michor and J. Slovak, "Natural Operations in Differential Geometry,", Springer-Verlag, (1993).   Google Scholar [6] B. S. Kruglikov and V. V. Lychagin, Geometry of differential equations,, in, (2008), 725.   Google Scholar [7] R. Ya. Matsyuk, Higher order variational origin of the Dixon's system and its relation to the quasi-classical 'Zitterbewegung' in general relativity,, preprint, ().   Google Scholar [8] J. Muñoz Masqué and I. M. Pozo Coronado, Parameter-invariant second-order variational problems in one varaiable,, J. Phys. A, 31 (1998), 6225.  doi: 10.1088/0305-4470/31/29/014.  Google Scholar [9] J. J. Stoker, "Differential Geometry,", Pure and Applied Mathematics, (1969).   Google Scholar
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