---
title: "Chaplygin Sleigh Interactive Lab | Sandbox Physics"
description: "It straightens out. The energy stays. Can turning fade without energy loss?"
canonical_url: "https://sandboxphysics.com/labs/chaplygin-sleigh/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/chaplygin-sleigh/"
last_modified: "2026-10-18"
---

Nonholonomic mechanics · Free motion

# Chaplygin Sleigh

Release a three-dimensional knife-edge sleigh. Move its center of mass, reverse the launch or put the mass over the blade. Compare the curved constrained path with a freely sliding center of mass.

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

Ideal planar rigid body with one lateral knife-edge constraint. No drag, blade friction loss, lift or tipping. Mass and inertia are specified independently of the schematic chassis.

Physics tutorial

## A constraint can redirect without dissipating

**Background** A knife edge allows motion along its length and forbids lateral motion. Its reaction acts sideways, so its power is zero.

**Why it matters** Connect a visible contact rule to the complete trajectory and quantitative account.

Start with the essentials

- **Focus question**: How can rotation fade away without losing mechanical energy?
- **One-sentence intuition**: The sideways constraint transfers energy between translation and rotation. For an offset mass center, the stable direction has the blade trailing the mass.

### Core mathematical model

#### Knife velocity and rigid-body dynamics

$$
\dot x=u\cos\theta,\quad\dot y=u\sin\theta,\quad\dot\theta=\omega,\qquad\dot u=a\omega^2,\quad\dot\omega=-\frac{a u\omega}{k^2+a^2}
$$

Coordinates refer to the knife contact location, not the mass center. Positive offset points forward along the blade; mass-center inertia equals mass times squared radius of gyration.

#### Energy and reaction

$$
E=\frac m2\left[u^2+(k^2+a^2)\omega^2\right],\quad N=\frac{m k^2 u\omega}{k^2+a^2},\quad P_N=Nv_{P,\perp}=0
$$

The mass center has a lateral velocity even though the blade does not. The reaction may be nonzero while doing no work.

### Common difficulties

#### Contact is a velocity rule

**Typical misconception** Returning a control or reducing yaw must undo motion or remove energy.

**Better mental model** Check the fixed energy total while its two terms exchange. The constraint does no work.

### Run the experiment

1. 01

   #### Experiment 1

   Release backward, then watch the forward speed reverse while energy stays constant.

   **What to observe:** Compare the synchronized path and numerical readouts.

2. 02

   #### Experiment 2

   Choose Mass over blade, then reverse the offset. Compare the circular and straightening paths.

   **What to observe:** Compare the synchronized path and numerical readouts.

3. 03

   #### Experiment 3

   Compare standard and fine integration at the largest yaw rate.

   **What to observe:** Compare the synchronized path and numerical readouts.

Physical conclusion

Put the mass center over the blade: the reaction no longer produces yaw torque, and a circular orbit can persist.

Primary theory: [Simoes & Colombo · rolling disk](https://arxiv.org/abs/2303.17920) · [Chang & Perlmutter · Chaplygin sleigh](https://arxiv.org/abs/1811.03757)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Measure gravity, split motion into components, add contact forces and collisions, then carry the same laws into orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

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