Helmholtz decomposition and potential functions for n-dimensional analytic vector fields

TL;DR

This paper generalizes Helmholtz decomposition to n-dimensional spaces by introducing a rotation potential, providing methods for decomposition, and demonstrating applications to complex dynamical systems.

Contribution

It introduces an n-dimensional rotation potential for Helmholtz decomposition, extending the method beyond three dimensions and offering multiple solution techniques.

Findings

Provides three methods for n-dimensional Helmholtz decomposition.

Derives closed-form solutions for various unbounded fields.

Includes applications to complex dynamical systems like Lorenz and Lotka-Volterra.

Abstract

The Helmholtz decomposition splits a sufficiently smooth vector field into a gradient field and a divergence-free rotation field. Existing decomposition methods impose constraints on the behavior of vector fields at infinity and require solving convolution integrals over the entire coordinate space. To allow a Helmholtz decomposition in $\mathbb{R}^n$, we replace the vector potential in $\mathbb{R}^3$ by the rotation potential, an n-dimensional, antisymmetric matrix-valued map describing $n(n-1)/2$ rotations within the coordinate planes. We provide three methods to derive the Helmholtz decomposition: (1) a numerical method for fields decaying at infinity by using an $n$-dimensional convolution integral, (2) closed-form solutions using line-integrals for several unboundedly growing fields including periodic and exponential functions, multivariate polynomials and their linear combinations, (3) an existence proof for all analytic vector fields. Examples include the Lorenz and R\"{o}ssler attractor and the competitive Lotka-Volterra equations with $n$ species.