Versal Normal Form for Nonsemisimple Singularities

TL;DR

This paper develops a systematic approach to versal normal form theory for nonsemisimple singularities, emphasizing the linear part first, and applies it to fluid dynamics and celestial mechanics problems.

Contribution

It provides a complete description of first order calculations for nilpotent cases and explicit transformations to versal normal form in specific applications.

Findings

Explicit first order calculations for 2D and 3D nilpotent cases

Transformation to versal normal form in fluid dynamics

Application to celestial mechanics L4 problem

Abstract

The theory of versal normal form has been playing a role in normal form since the introduction of the concept by V.I. Arnol'd. But there has been no systematic use of it that is in line with the semidirect character of the group of formal transformations on formal vector fields, that is, the linear part should be done completely first, before one computes the nonlinear terms. In this paper we address this issue by giving a complete description of a first order calculation in the case of the two- and three-dimensional irreducible nilpotent cases, which is then followed up by an explicit almost symplectic calculation to find the transformation to versal normal form in a particular fluid dynamics problem and in the celestial mechanics \(L_4\) problem.