---
title: "Double-Well, Barrier & Separatrix Lab Interactive Lab | Sandbox Physics"
description: "How does a barrier between two wells change the way a particle moves? It depends on whether the energy clears the barrier. Below the top, the particle rocks back and forth in one well; above it, the particle runs between both wells. Close to the dividing energy it crawls past the top of the barrier, so the period becomes very long. Damping can drain just enough energy to trap a particle that used to cross."
canonical_url: "https://sandboxphysics.com/labs/double-well/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/double-well/"
last_modified: "2026-10-22"
---

M045 · Barriers and critical slowing

# Double-Well, Barrier & Separatrix Lab

**How does a barrier between two wells change the way a particle moves?** It depends on whether the energy clears the barrier. Below the top, the particle rocks back and forth in one well; above it, the particle runs between both wells. Close to the dividing energy it crawls past the top of the barrier, so the period becomes very long. Damping can drain just enough energy to trap a particle that used to cross.

Set the barrier, the well spacing, the tilt and the damping. Compare a particle trapped in one well with one that crosses between wells, approach the separatrix from below and from above, and let damping capture a fast particle on one side. Measure crossings and periods, and check the periods against an independent calculation from the energy integral.

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

The model is an ideal point particle moving along a horizontal guide in a given polynomial potential, with all quantities in SI units. The 3D force probes and the bands that mark where the particle can go are drawn guides; the Lab does not simulate rolling, a physical barrier or any actuator. The vertical curve is a graph of energy, not a real track. Gray contours are drawn at the release energy, and the recorded motion is teal. Playback recomputes the motion from saved checkpoints. The step-halving comparison covers at most the first ten seconds; the energy check covers every integration step.

Physics tutorial

## A separatrix separates confined and crossing orbits

**Background** A quartic double well has two stable minima with a saddle, the top of the barrier, between them. Without damping, a particle with less energy than the saddle stays on one side; with more, it can reach both wells. At exactly the saddle energy, a moving particle creeps ever closer to the saddle and never completes a cycle. That orbit is the separatrix.

**Why it matters** The energy and phase-space picture is standard classical mechanics for motion along one coordinate. The barrier height and the largest allowed tilt are worked out directly from the quartic potential shown. With viscous damping, the model counts every bit of kinetic energy the damping removes.

Start with the essentials

- **Focus question**: Why does crossing a little above the barrier take so much longer?
- **One-sentence intuition**: As the energy approaches the top of the barrier, the particle lingers longer and longer near the saddle. Damping can drain enough energy that an orbit that used to cross gets trapped on one side.

### Core mathematical model

#### Tilted double well

$$
U(x)=B\left[\left(\frac{x}{a}\right)^2-1\right]^2+fx
$$

Without tilt, the barrier scale is the height of the barrier above either minimum. With tilt, the barrier shown for the release side is recomputed from where the minima and the saddle actually are.

#### When both minima survive

$$
|f|<f_{\mathrm{crit}}=\frac{8B}{3\sqrt3\,a},\qquad U\prime(x_*)=0
$$

The tilt is kept below the saddle-node threshold, so the two minima and the saddle between them always stay distinct.

#### Energy balance with viscous damping

$$
m\ddot x=-U\prime(x)-c\dot x,\qquad H(t)+Q(t)=H(0),\qquad Q(t)=\int_0^t c\dot x^2\,dt
$$

Each damping substep shrinks the velocity exponentially and adds the exact kinetic energy it removes to the heat. Errors from the conservative part of the step still show up as a mismatch in the balance.

#### Energy above the barrier and the undamped period

$$
\Delta H=H-U(x_s),\qquad T(H)=\sqrt{2m}\int_{x_-}^{x_+}\frac{dx}{\sqrt{H-U(x)}}
$$

The orbit at exactly the critical energy has no finite period, so the plot leaves a gap between the branches below and above the barrier.

#### Motion near the saddle

$$
U(x)\simeq U(x_s)-\frac{m\lambda_s^2}{2}(x-x_s)^2,\qquad \lambda_s^2=-\frac{U\prime\prime(x_s)}{m}
$$

The period grows logarithmically near the saddle, which magnifies small energy errors. A particle released exactly at the saddle stays at rest unless something nudges it.

### Common difficulties

#### Crossings at the critical energy come from the numerics

**Typical misconception** Every crossing recorded at the critical energy really happens.

**Better mental model** The exact separatrix does not cross the saddle in any finite time. Error from a finite time step can tip it into crossing or turning back, so compare the energy error with the half-step difference.

#### With tilt, the barrier scale is not the real barrier

**Typical misconception** The symmetric barrier setting always equals the barrier height on the launch side.

**Better mental model** Tilt shifts the energies of both minima and of the saddle. The readout uses the actual difference between them.

#### Damping changes the orbit

**Typical misconception** After the particle loses energy, the measured period still equals the undamped reference period.

**Better mental model** As energy drains away, the orbit keeps changing. Capture is reported only once the recorded energy sits below the saddle by more than the numerical uncertainty.

### Run the experiment

1. 01

   #### Compare trapped and crossing motion

   Acquire full records for Below the barrier and Across both wells, then look at the phase portrait and the recorded saddle crossings.

   **What to observe:** Without damping, only motion above the barrier reaches both wells.

2. 02

   #### Approach the separatrix

   Compare Approach from below with Approach from above, then select Critical energy.

   **What to observe:** Both periods get longer. At the critical energy the reference shows no finite period, and the result becomes much more sensitive to numerical error.

3. 03

   #### Let damping capture the particle

   Acquire a full record with Dissipative capture and look at heat plus mechanical energy.

   **What to observe:** Heat plus mechanical energy stays equal to the starting energy, and the orbit ends up trapped on one side.

4. 04

   #### Tilt without erasing a well

   Change the tilt and drag the release point in phase space. Compare the actual saddle energy with the symmetric barrier scale.

   **What to observe:** The saddle shifts and the barrier is higher from one side than from the other, yet both minima remain.

Physical conclusion

Near the barrier the period grows without bound and the result grows sensitive to numerical error, so read the two together.

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)