Title: | Homogeneous systems of higher-order ordinary differential equations (English) |

Author: | Crampin, Mike |

Language: | English |

Journal: | Communications in Mathematics |

ISSN: | 1804-1388 |

Volume: | 18 |

Issue: | 1 |

Year: | 2010 |

Pages: | 37-50 |

Summary lang: | English |

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Category: | math |

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Summary: | 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. (English) |

MSC: | 34A26 |

MSC: | 34C14 |

MSC: | 53A55 |

MSC: | 53B15 |

MSC: | 83C80 |

idZBL: | Zbl 1244.34010 |

idMR: | MR2848505 |

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Date available: | 2011-10-25T07:17:18Z |

Last updated: | 2013-10-22 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/141671 |

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Reference: | [7] Crampin, M., Saunders, D.J.: Affine and projective transformations of Berwald connections.Diff. Geom. Appl. 25 2007 235–250 Zbl 1158.53055, MR 2330452, 10.1016/j.difgeo.2007.02.001 |

Reference: | [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 |

Reference: | [9] Fritelli, S., Kozameh, C., Newman, E.T.: Differential geometry from differential equations.Comm. Math. Phys. 223 2001 383–408 MR 1864438, 10.1007/s002200100548 |

Reference: | [10] Godliński, M., Nurowski, P.: Third order ODEs and four-dimensional split signature Einstein metrics.J. Geom. Phys. 56 2006 344–357 MR 2171889, 10.1016/j.geomphys.2005.01.011 |

Reference: | [11] Godliński, M., Nurowski, P.: Geometry of third-order ODEs.preprint: arXiv:0902.4129v1 [math.DG] |

Reference: | [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 Zbl 1048.34019, MR 1893906, 10.1016/S0926-2245(02)00065-7 |

Reference: | [13] Shen, Z.: Differential Geometry of Spray and Finsler Spaces.Kluwer 2001 Zbl 1009.53004, MR 1967666 |

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