Homogeneity and projective equivalence of differential equation fields

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

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, 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

Abstract
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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.

Keywords: reparametrization., homogeneity, Ordinary differential equations, parametric Lagrangian, projective equivalence.

Mathematics Subject Classification: 34A26, 70H03, 70H5.

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), 1291-1327. doi: 10.1142/S0219887811005701.
[2]	M. Crampin, Homogeneous systems of higher-order ordinary differential equations, Communications in Mathematics, 18 (2010), 37-50.
[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), 657-689.
[4]	F. Faà di Bruno, Sullo sviluppo delle Funzioni, Annali di Scienze Matematiche e Fisiche, 6 (1855), 479-480.
[5]	I. Kolář, P. W. Michor and J. Slovak, "Natural Operations in Differential Geometry," Springer-Verlag, Berlin, 1993.
[6]	B. S. Kruglikov and V. V. Lychagin, Geometry of differential equations, in "Handbook of Global Analysis" (eds. D. Krupka and D. J. Saunders), 1214, Elsevier Sci. B. V., Amsterdam, (2008), 725-771.
[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, arXiv:1101.5384.
[8]	J. Muñoz Masqué and I. M. Pozo Coronado, Parameter-invariant second-order variational problems in one varaiable, J. Phys. A, 31 (1998), 6225-6242. doi: 10.1088/0305-4470/31/29/014.
[9]	J. J. Stoker, "Differential Geometry," Pure and Applied Mathematics, Vol. XX, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1969.

show all references

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), 1291-1327. doi: 10.1142/S0219887811005701.
[2]	M. Crampin, Homogeneous systems of higher-order ordinary differential equations, Communications in Mathematics, 18 (2010), 37-50.
[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), 657-689.
[4]	F. Faà di Bruno, Sullo sviluppo delle Funzioni, Annali di Scienze Matematiche e Fisiche, 6 (1855), 479-480.
[5]	I. Kolář, P. W. Michor and J. Slovak, "Natural Operations in Differential Geometry," Springer-Verlag, Berlin, 1993.
[6]	B. S. Kruglikov and V. V. Lychagin, Geometry of differential equations, in "Handbook of Global Analysis" (eds. D. Krupka and D. J. Saunders), 1214, Elsevier Sci. B. V., Amsterdam, (2008), 725-771.
[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, arXiv:1101.5384.
[8]	J. Muñoz Masqué and I. M. Pozo Coronado, Parameter-invariant second-order variational problems in one varaiable, J. Phys. A, 31 (1998), 6225-6242. doi: 10.1088/0305-4470/31/29/014.
[9]	J. J. Stoker, "Differential Geometry," Pure and Applied Mathematics, Vol. XX, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1969.

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