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The concept of homogeneity, which picks out sprays from the general run of systems of second-order ordinary differential equations in the geometrical theory of such equations, is generalized so as to apply to equations of higher order. Certain properties of the geometric concomitants of a spray are shown to continue to hold for higher-order systems. Third-order equations play a special role, because a strong form of homogeneity may apply to them. The key example of a single third-order equation which is strongly homogeneous in this sense states that the Schwarzian derivative of the dependent variable vanishes. This equation is of importance in the theory of the association between third-order equations and pseudo-Riemannian manifolds due to Newman and his co-workers.
[1] Anderson, I., Thompson, G.: The inverse problem of the calculus of variations for ordinary differential equations. Mem. Amer. Math. Soc. 98 1992 No. 473 MR 1115829 | Zbl 0760.49021
[2] de Andrés, L.C., de León, M., Rodrigues, P.R.: Canonical connections associated with regular Lagrangians of higher order. Geom. Dedicata 39 1991 17–28 MR 1116206
[3] Antonelli, P., Bucataru, I.: KCC theory of a system of second order differential equations. Handbook of Finsler Geometry Vol. 1 , Antonelli (ed.)Kluwer 2003 83–174 MR 2066445 | Zbl 1105.53017
[4] Bucataru, I., Constantinescu, O., Dahl, M.F.: A geometric setting for systems of ordinary differential equations. preprint: arXiv:1011.5799 [math.DG]
[5] Crampin, M.: Connections of Berwald type. Publ. Math. Debrecen 57 2000 455–473 MR 1798727 | Zbl 0980.53031
[6] Crampin, M., Sarlet, W., Cantrijn, F.: Higher-order differential equations and higher-order Lagrangian mechanics. Math. Proc. Cam. Phil. Soc. 99 1986 565–587 DOI 10.1017/S0305004100064501 | MR 0830369 | Zbl 0609.58049
[7] Crampin, M., Saunders, D.J.: Affine and projective transformations of Berwald connections. Diff. Geom. Appl. 25 2007 235–250 DOI 10.1016/j.difgeo.2007.02.001 | MR 2330452 | Zbl 1158.53055
[8] Crampin, M., Saunders, D.J.: On the geometry of higher-order ordinary differential equations and the Wuenschmann invariant. Groups, Geometry and Physics , Clemente-Gallardo and Martínez (eds.)Monografía 29, Real Academia de Ciencias de Zaragoza 2007 79–92 MR 2288307
[9] Fritelli, S., Kozameh, C., Newman, E.T.: Differential geometry from differential equations. Comm. Math. Phys. 223 2001 383–408 DOI 10.1007/s002200100548 | MR 1864438
[10] Godliński, M., Nurowski, P.: Third order ODEs and four-dimensional split signature Einstein metrics. J. Geom. Phys. 56 2006 344–357 DOI 10.1016/j.geomphys.2005.01.011 | MR 2171889
[11] Godliński, M., Nurowski, P.: Geometry of third-order ODEs. preprint: arXiv:0902.4129v1 [math.DG]
[12] Saunders, D.J.: On the inverse problem for even-order ordinary differential equations in the higher-order calculus of variations. Diff. Geom. Appl. 16 2002 149–166 DOI 10.1016/S0926-2245(02)00065-7 | MR 1893906 | Zbl 1048.34019
[13] Shen, Z.: Differential Geometry of Spray and Finsler Spaces. Kluwer 2001 MR 1967666 | Zbl 1009.53004
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