---
title: "How Signed Singularities Assemble Fluid Orbits Interactive Lab | Sandbox Physics"
description: "How do a handful of point vortices move one another? Each vortex simply drifts with the flow that all the others create."
canonical_url: "https://sandboxphysics.com/labs/point-vortex-dynamics/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/point-vortex-dynamics/"
last_modified: "2026-10-27"
---

Fluid dynamics · Hamiltonian vortex motion

# Point-Vortex Dynamics Studio

**How do a handful of point vortices move one another?** Each vortex simply drifts with the flow that all the others create. Two vortices of the same sign circle a common center; two of opposite sign travel together. Two such opposite-sign pairs can leapfrog through each other, and with four or more vortices, partners can swap chaotically.

Drag point vortices around a two-dimensional phase plane and compare a corotating pair, a translating dipole, leapfrogging dipoles and four interacting vortices. The velocity field, the tracers, the minimum distance between vortices and the Hamiltonian drift all update together.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

This two-dimensional inviscid model moves every vortex with the velocity $\dot{\mathbf x}_i$ induced by all the others. The core scale $\varepsilon$ removes the numerical singularity but changes close encounters.

Physics tutorial

## Point vortices: complex flow from minimal degrees of freedom

**Background** In a two-dimensional inviscid flow, a small, concentrated patch of vorticity looks like a point from far away, described only by its position and circulation. A point vortex does not move itself. It is carried along by the velocity that all the other vortices induce through the Biot–Savart law.

**Why it matters** The model is a classic workbench for translating dipoles, corotating pairs, leapfrogging dipoles, and few-body chaos. Aref and collaborators described it in their 2007 review as a classical mathematics playground connecting fluids, Hamiltonian mechanics, and dynamical systems.

Start with the essentials

- **Focus question**: How can nothing more than positions and signed circulations produce stable orbits and complicated exchanges?
- **One-sentence intuition**: Vortex $i$ moves with the sum of the velocities induced by every other vortex. Two vortices of the same sign circle a common center; two of opposite sign travel together at right angles to the line joining them.

### Core mathematical model

#### Soft-core point-vortex equations

$$
\dot x_i=-\frac{1}{2\pi}\sum_{j\ne i}\Gamma_j\frac{y_i-y_j}{r_{ij}^2+\varepsilon^2},\qquad \dot y_i=\frac{1}{2\pi}\sum_{j\ne i}\Gamma_j\frac{x_i-x_j}{r_{ij}^2+\varepsilon^2}
$$

The core scale removes the singularity at zero distance; outside the core, the classical point-vortex velocity returns.

#### Regularized Hamiltonian

$$
H=-\frac{1}{4\pi}\sum_{i<j}\Gamma_i\Gamma_j\ln\!\left(r_{ij}^2+\varepsilon^2\right)
$$

With no background strain, and away from jumps at the periodic boundary, it is conserved and gives a sensitive check on the numerical integration.

#### Exact speeds for two softened vortices

$$
\omega_{++}=\frac{\Gamma}{\pi(d^2+\varepsilon^2)},\qquad U_{+-}=\frac{|\Gamma|d}{2\pi(d^2+\varepsilon^2)}
$$

Two equal vortices of the same sign rotate at this rate; two equal vortices of opposite sign form a dipole that moves at this speed.

### Common difficulties

#### A vortex does not push itself along

**Typical misconception** The visible spin of a point vortex gives that vortex its own forward velocity.

**Better mental model** With symmetric softening, the velocity a vortex induces on itself is zero. Its motion comes entirely from the other vortices and the background flow.

#### Softening is not viscous diffusion

**Typical misconception** A larger softening core is the same as a more viscous fluid.

**Better mental model** Softening only smooths the near-field singularity; it does not spread the core in time or dissipate circulation.

### Run the experiment

1. 01

   #### Check a corotating pair

   Choose Corotating pair with zero background strain and watch pair distance and Hamiltonian drift.

   **What to observe:** The pair rotates about its midpoint, the separation stays nearly constant and the numerical drift stays small.

2. 02

   #### Reverse one circulation

   Switch to Translating dipole and compare the trails with passive tracers.

   **What to observe:** The two vortices keep their separation and travel together, carrying the surrounding fluid along in a dipole pattern.

3. 03

   #### Break the integrable symmetry

   Choose Leapfrogging pairs or Perturbed quadrupole, then drag one core.

   **What to observe:** Which vortices swap places now depends sensitively on the starting layout. The conserved quantities still tell real sensitivity apart from numerical error.

Physical conclusion

With very few degrees of freedom, the point-vortex model keeps the essential geometry of induced motion: the signs of the circulations shape the orbits, and the Hamiltonian and impulse check whether the numerical orbit can be trusted.

Conceptual inspiration: [FluidMotion](http://www.bugman123.com/FluidMotion/index.html). Original simulations and graphics; the model limits are described above.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)