Point Vortex Dynamics on Closed Surfaces

2026-02-03

AI Summary

This work addresses the challenge of effectively generalizing point vortex dynamics to arbitrary closed surfaces while preserving both computational efficiency and physical consistency. Focusing on closed surfaces of genus zero with zero total vorticity, the study establishes a unified theoretical framework that seamlessly integrates differential geometry, fluid dynamics, and Hamiltonian mechanics. This framework systematically unifies point vortex models previously developed for the plane, the sphere, and general curved surfaces. Beyond providing rigorous theoretical derivations, the paper also offers practical guidance for numerical implementation, thereby laying the foundation for efficient, structure-preserving simulations of vortex flows on curved manifolds. The approach significantly extends the applicability of classical point vortex theory to a broad class of non-Euclidean geometries.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Low Level & Physics-based Vision
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

The theory of point vortex dynamics has existed since Kirchhoff's proposal in 1891 and is still under development with connections to many fields in mathematics. As a strong simplification of the concept of vorticity it excels in computational speed for vorticity based fluid simulations at the cost of accuracy. Recent finding by Stefanella Boatto and Jair Koiller allowed the extension of this theory on to closed surfaces. A comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity is presented here. Additionally fundamental knowledge of fluid dynamics and surfaces are explained in a way to unify the theory of point vortex dynamics of the plane, the sphere and closed surfaces together with implementation details and supplement material.

Problem

• point vortex dynamics
• closed surfaces
• vorticity
• fluid dynamics
• genus zero

Innovation

• point vortex dynamics
• closed surfaces
• fluid simulation
• vorticity
• geometric fluid dynamics