---
abstract: 'We describe a method for computing Casimir invariants that is applicable to both finite and infinite-dimensional Poisson brackets. We apply the method to various finite and infinite-dimensional examples, including a Poisson bracket embodying both finite and infinite-dimensional structure.'
author:
- 'T. W. Yudichak$^{\;(1)}$'
- 'Benito Hernández–Bermejo$^{\;(2)}$'
- 'P. J. Morrison$^{\;(1,*)}$'
title: Computing Casimir invariants from Pffafian systems
---

6.5in 0.0in 0.0in -5in -.5in 0.0in

[*$^1$ Department of Physics and Institute for Fusion Studies, The University of Texas at Austin, Austin, TX 78712, USA*]{}

[*$^2$Departamento de Física Matemática y Fluidos, Universidad Nacional de Educación a Distancia. Senda del Rey S/N, 28040 Madrid, Spain. E-mail: bhernand@apphys.uned.es*]{}

[*Keywords:*]{} Poisson systems, Casimir functions, noncanonical brackets, Frobenius theorem, Pfaffian systems.

[*PACS:*]{} 03.20.+i, 03.40.-t, 02.40.+m, 51.10.+y.

$^*$ To whom all correspondence should be addressed.

Introduction {#S-INTRO}
============

Noncanonical Hamiltonian, or Poisson, systems have been studied extensively by mathematical physicists, and with good reason. They arise in fields as diverse as dynamical systems theory [@gyn1; @pla1], fluid dynamics and plasma physics [@flpl; @olv3], and condensed matter physics [@cmph]. (For more references see [@mara; @mrmp].)

One of the key features of Hamiltonian systems is the existence of a Poisson bracket. This object finds many applications, including 