---
title: "Lagrange Points & Three-Body Channels Interactive Lab | Sandbox Physics"
description: "If a probe has enough energy to cross from one body's region to the other's, will it actually cross?"
canonical_url: "https://sandboxphysics.com/labs/lagrange-channels/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/lagrange-channels/"
last_modified: "2026-11-04"
---

M076 · A moving balance

# Lagrange Points & Three-Body Channels

[Restricted three-body orbit](https://sandboxphysics.com/labs/lagrange-channels/) [Binary mass transfer](https://sandboxphysics.com/labs/lagrange-channels/?analysis=mass-transfer)

**If a probe has enough energy to cross from one body's region to the other's, will it actually cross?** Not necessarily; enough energy only makes the crossing possible. At that energy, the region the probe cannot enter opens a narrow neck between the two areas. Whether the probe actually passes through depends on where its full trajectory takes it.

Place a massless probe near two bodies that orbit each other. Launch it near a Lagrange point, open the narrow neck in the zero-velocity boundary, and compare the same path in the rotating and inertial frames.

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

Model assumptions

This is the planar, circular restricted three-body model. It does not compute halo orbits and is not a tool for mission navigation.

Physics tutorial

## Lagrange Points & Three-Body Channels

**Background** In a frame that rotates with the two bodies, the balance points of the orbit stand still.

**Why it matters** The region a probe is allowed to enter is not the same as the path it actually takes, and learning to tell them apart is the point.

Start with the essentials

- **Focus question**: Why are some Lagrange points stable and others unstable?
- **One-sentence intuition**: An open neck means the probe is allowed to pass through, not that it will.

### Core mathematical model

#### Equations of motion in the rotating frame

$$
\ddot x-2\dot y=\partial_x\Omega,\quad\ddot y+2\dot x=\partial_y\Omega
$$

The Coriolis terms depend on the probe's velocity.

#### Effective potential

$$
\Omega=\frac{x^2+y^2}{2}+\frac{1-\mu}{r_1}+\frac{\mu}{r_2}
$$

In this rotating frame the two main bodies stay fixed.

#### Where the probe can go

$$
C=2\Omega-\dot x^2-\dot y^2,\quad 2\Omega\geq C
$$

Entering a forbidden region would require the speed squared to be negative, which is impossible.

#### Stability condition for the triangular points

$$
27\mu(1-\mu)<1
$$

This tests linear stability in the plane only. It does not guarantee what larger orbits will do.

### Common difficulties

#### Gravity at a Lagrange point

**Typical misconception** There is no gravity at a Lagrange point.

**Better mental model** Gravity is still there; it supplies the acceleration the orbit needs. The balance with centrifugal force appears only in the rotating frame.

### Run the experiment

1. 01

   #### Place a probe

   Drag the gold start and green velocity handles.

   **What to observe:** The Jacobi constant and the forbidden region change together.

2. 02

   #### Release near a balance point

   Compare Near the triangular point with Nudge the saddle.

   **What to observe:** A probe that starts close to a balance point may stay nearby or leave quickly.

3. 03

   #### Change the frame

   Pause, switch frames and replay.

   **What to observe:** The same recorded trajectory looks different in each frame.

Physical conclusion

Always read a path together with the frame it is drawn in and its Jacobi constant.

## Related experiments and articles

Binary mass transfer and common envelopes · Reading path

### Binary mass transfer and common envelopes

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **When one star spills onto another**](https://sandboxphysics.com/stories/binary-mass-transfer/when-a-star-spills/)
2. [Article **Finding the passage between two stars**](https://sandboxphysics.com/stories/binary-mass-transfer/the-passage-at-l1/)
3. [Article **Why exchanging mass can widen an orbit**](https://sandboxphysics.com/stories/binary-mass-transfer/exchanging-mass-changing-orbits/)
4. [Article **When mass transfer feeds its own overflow**](https://sandboxphysics.com/stories/binary-mass-transfer/when-transfer-feeds-itself/)
5. [Article **A changing period in Beta Lyrae**](https://sandboxphysics.com/stories/binary-mass-transfer/beta-lyrae-period-growth/)
6. [Article **Where common-envelope energy goes**](https://sandboxphysics.com/stories/binary-mass-transfer/where-common-envelope-energy-goes/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an 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)