A classification of volume preserving generating forms in R^3

TL;DR

This paper classifies all volume preserving generating forms in three-dimensional space, identifying five distinct classes, including two new ones, and connects them to volume preserving numerical schemes.

Contribution

It reduces the 36 known forms to five fundamental classes and links these to volume preserving integrators, including the discovery of two novel classes.

Findings

Reduced 36 cases to 5 fundamental classes.

Identified 2 new classes of generating forms.

Connected forms to volume preserving numerical schemes.

Abstract

In earlier work, Lomeli and Meiss used a generalization of the symplectic approach to study volume preserving generating differential forms. In particular, for the $\mathbb{R}^3$ case, the first to differ from the symplectic case, they derived thirty-six one-forms that generate exact volume preserving maps. Xue and Zanna had studied these differential forms in connection with the numerical solution of divergence-free differential equations: can such forms be used to devise new volume preserving integrators or to further understand existing ones? As a partial answer to this question, Xue and Zanna showed how six of the generating volume form were naturally associated to consistent, first order, volume preserving numerical integrators. In this paper, we investigate and classify the remaining cases. The main result is the reduction of the thirty-six cases to five essentially different cases, up to variable relabeling and adjunction. We classify these five cases, identifying two novel classes and associating the other three to volume preserving vector fields under a Hamiltonian or Lagrangian representation. We demonstrate how these generating form lead to consistent volume preserving schemes for volume preserving vector fields in $\mathbb{R}^3$.