2011/09/16 by M. Crampin, Crampin, M., D. J. Saunders +2
Computer Science · Engineering · Mathematics · #34A26 #70H03 #70H50 #Advanced Numerical Analysis Techniques #Advanced Vision and Imaging #Differential Geometry (math.DG) #FOS: Mathematics #Polynomial and algebraic computation #math.DG #msc:34A26 #msc:70H03 #msc:70H50
paper · pdf · doi:10.48550/arxiv.1109.3640
arxiv created 2011/09/16 · openalex publication_date 2011/09/16 · arxiv updated 2011/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.