Generalized Normal Forms of Infinitesimal Symplectic and Contact Transformations in the Neighborhood of a Singular Point

TL;DR

This paper introduces a generalized normal form for systems of ODEs near singular points, applicable to symplectic and contact transformations, with practical examples in plasma physics and non-ideal gases.

Contribution

It presents a new method for deriving generalized normal forms for infinitesimal symplectic and contact transformations near singular points, including an application to physical systems.

Findings

Normal forms for equations of state of non-ideal gases and plasmas are derived.

Lowest order perturbation effects in Debye-Hückel hydrogen plasmas are characterized by three parameters.

The method facilitates studying critical phenomena in non-ideal media.

Abstract

Definition of generalized normal form for a system of ODEs corresponding to an infinitesimal symplectic or contact transformation near a singular point, with an arbitrary polynomial unperturbed part, and a method of its finding are introduced. Applicability of the introduced method to studying the critical phenomena in non-ideal media is shown. As examples, generalized normal forms for the equations of state of a mixture of non-ideal gases and non-ideal multicomponent plasma are considered within the framework of perturbation theory. In particular, it is shown that the lowest order perturbation effects in the Debye-H\"uckel hydrogen plasmas are classified by only three constant parameters.