Symmetries and Weighted Integrability of Vector Fields with Jacobi Multipliers

TL;DR

This work extends KAM-type persistence to divergence-free vector fields and to vector fields with Jacobi multipliers on $n$-manifolds, introducing quantitative measures of partial integrability via the unweighted functional $rak m$ and its weighted counterpart $m_ ho$. It establishes continuity results at Arnold-integrable nondegenerate systems and derives structural rigidity for symmetries in the Jacobi-multiplier setting, showing symmetries are tangent to invariant tori under appropriate nondegeneracy/resonance. A practical numerical scheme based on finite-time Lyapunov exponents is developed to compute $m_ ho(V)$, enabling exploration of integrable, partially integrable, and irregular regimes; demonstrated on a nonlinear weighted example where the nonlinear parameter $oldsymbol{ m α}$ drives the transition across regimes. This work provides both analytic and computational tools to quantify and visualize the persistence of invariant structures under weighted volume preservation, with potential applications in fluid dynamics, magnetohydrodynamics, and weighted Hamiltonian-like 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 $C^ω$ 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 $dμ_ρ=ρ\,Ω$. 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} $m_ρ(V)$, measuring the weighted fraction of phase space occupied by quasi-integrable invariant tori of a vector field satisfying $\operatorname{div}(ρV)=0$. Finally, we develop a numerical algorithm based on finite-time Lyapunov exponents to compute $m_ρ(V)$, 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 $α$ increases.

Figures (3)

• Figure : 3D projection of a trajectory $(x_1,y_1,x_2)$ for the weighted nonlinear divergence-free system $V = \frac{1}{\rho}(Lu + \alpha N(u)), \qquad \rho = 1 + \varepsilon(x_1^2 + y_1^2 + x_2^2 + y_2^2),$with parameters $\varepsilon=0.5$, $\delta = 0.3$, and $\alpha = 0$. In this exactly integrable regime, all trajectories lie on smooth weighted invariant tori, corresponding to quasi-periodic motion with no torus deformation. The weighted partial integrability functional reaches its maximal value $m_\rho(V)=1$, indicating that the entire weighted phase space is foliated by invariant tori.
• Figure : Numerical simulation of the weighted nonlinear divergence-free system in the intermediate regime ($m_\rho(V)\approx 0.69$). The 3D projection of the trajectory $(x_1,y_1,x_2)$ shows a deformed, quasi-toroidal structure: the orbit remains confined within a bounded region and exhibits slow phase mixing, yet lacks the perfect periodicity of the integrable case. This illustrates a partially integrable regime in which invariant tori persist only in a weighted sense under the nonuniform density $\rho=1+\varepsilon(x_1^2+y_1^2+x_2^2+y_2^2)$. The trajectory does not fill the entire domain, indicating residual coherence, but it no longer closes upon itself, marking the onset of quasi-ergodic behavior.
• Figure : 3D projection of a trajectory $(x_1,y_1,x_2)$ for the weighted nonlinear divergence-free system $V = \frac{1}{\rho}(Lu + \alpha N(u)), \qquad \rho = 1 + \varepsilon(x_1^2 + y_1^2 + x_2^2 + y_2^2),$with parameters $\varepsilon = 0.5$, $\delta = 0.3$, and $\alpha = 0.5$. The orbit is no longer confined to a smooth invariant torus but fills a thickened region of phase space, indicating the onset of quasi-ergodic behavior. This corresponds to the weighted partially integrable regime, in which some invariant tori persist while others are destroyed. As $\alpha$ increases, nonlinear effects distort the angular frequencies, leading to resonance overlap and progressive breakup of tori, and consequently to a decrease of the weighted integrability functional $m_{\rho}(V)$.

Theorems & Definitions (7)

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