Normal Forms of second order Ordinary Differential Equations $y_{xx}=J(x,y,y_{x})$ under FibrePreserving Maps
Abstract
We study the equivalence problem of classifying second order ordinary differential equations $y_{xx}=J(x,y,y_{x})$ modulo fibrepreserving point transformations $x\longmapsto \varphi(x)$, $y\longmapsto \psi(x,y)$ by using Moser's method of normal forms. We first compute a basis of the Lie algebra ${\frak{g}}_{{\{y_{xx}=0\}}}$ of fibrepreserving symmetries of $y_{xx}=0$. In the formal theory of Moser's method, this Lie algebra is used to give an explicit description of the set of normal forms $\mathcal{N}$, and we show that the set is an ideal in the space of formal power series. We then show the existence of the normal forms by studying flows of suitable vector fields with appropriate corrections by the CauchyKovalevskaya theorem. As an application, we show how normal forms can be used to prove that the identical vanishing of HsuKamran primary invariants directly imply that the second order differential equation is fibrepreserving point equivalent to $y_{xx}=0$.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 DOI:
 10.48550/arXiv.2109.02107
 arXiv:
 arXiv:2109.02107
 Bibcode:
 2021arXiv210902107F
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Complex Variables;
 Mathematics  Dynamical Systems
 EPrint:
 27 pages