Symmetries and Weighted Integrability of Vector Fields with Jacobi Multipliers

TL;DR

This paper explores the structure and stability of divergence-free and Jacobi multiplier vector fields on Riemannian manifolds, introducing new functionals to measure invariant tori and developing numerical methods to analyze their phase space behavior.

Contribution

It introduces the weighted partial integrability functional for vector fields with Jacobi multipliers and provides a numerical algorithm to evaluate phase space integrability.

Findings

Functional is continuous at analytic, nondegenerate divergence-free fields.

Persistence and breakdown of invariant tori under perturbations analyzed.

Numerical algorithm effectively measures weighted invariant tori in weighted systems.

Abstract

In this paper, we investigate analytic divergence-free vector fields and vector fields admitting a Jacobi multiplier on $n$-dimensional Riemannian manifolds. We first introduce a functional acting on the space of divergence-free vector fields that quantifies the fraction of the manifold foliated by ergodic invariant tori, and establish a Kolmogorov--Arnold--Moser (KAM) type theorem for such systems. We prove that this functional is continuous at analytic, nondegenerate, Arnold--integrable divergence-free vector fields with respect to the $C^{\omega}$ topology, and analyze the persistence and breakdown of invariant $(n-1)$-dimensional tori under small perturbations. Extending this framework, we study vector fields possessing Jacobi multipliers, which generalize divergence-free fields by preserving a weighted volume form $d\mu_{\rho}=\rho\,\Omega$. We derive the local structure of their symmetry fields and show that, under suitable nondegeneracy and resonance conditions, every symmetry must be tangent to the invariant tori. We then define the \emph{weighted partial integrability functional} $m_{\rho}(V)$, measuring the weighted fraction of phase space occupied by quasi-integrable invariant tori of a vector field satisfying $\operatorname{div}(\rho V)=0$. Finally, we develop a numerical algorithm based on finite-time Lyapunov exponents to compute $m_{\rho}(V)$, and illustrate its behavior on a weighted nonlinear divergence-free system, showing the transition from integrable to partially integrable and irregular regimes as the nonlinear parameter $\alpha$ increases.