Article Contents
Article Contents

# Homogeneity and projective equivalence of differential equation fields

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

 Citation:

•  [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.