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.

Give the particle just enough energy to cross a barrier. Compare trapping, slow passage near the saddle and damping that eventually holds the particle in one well.

Particle position $0$

Velocity $0$

Mechanical energy $0$

Energy above saddle $0$

Selected time $0$

Conservative force $0$

Viscous force $0$

Potential energy $0$

Kinetic energy $0$

Dissipated energy $0$

Local small-oscillation rate $0$

Selected minimum $0$

Initial left energy boundary $0$

Initial right energy boundary $0$

Conservative energy-integral period $0$

Maximum energy balance error $0$

Half-step position difference $0$

Recorded saddle crossings $0$

Period quadrature difference $0$

Release-side barrier height $0$

Saddle energy $0$

Saddle position $0$

Enough energy to cross?

Symmetric barrier scale

B

1\,\mathrm{J}

Launch energy / barrier scale

K_0/B

1.08

Viscous damping

c

0\,\mathrm{kg\,s^{-1}}

Well asymmetry

f/f_{\mathrm{crit}}

0

Half well spacing

a

1\,\mathrm{m}

Launch direction

s_v

1

Release position

x_0

-1\,\mathrm{m}

Particle mass

m

1\,\mathrm{kg}

Record duration

t_{\mathrm{record}}

40\,\mathrm{s}

Requested time step

h

5\,\mathrm{ms}

Selected time

t

3.2\,\mathrm{s}

Drag the particle to set its release position; the arrow keys do the same. Energy and phase plots share the same solved trajectory.

SI units; rightward is positive. An ideal point mass follows a prescribed quartic double-well force with optional linear viscous damping. The guide and sphere are schematic; no rolling or actuator mechanism is modelled. The energy curve is a graph, not a physical track. Gray phase contours are conservative references at release energy, even when the numerical trajectory loses energy. The barrier setting is the symmetric scale; the actual release-side barrier changes with tilt. Energy plus heat is audited at every integration step; the half-step position comparison covers at most ten seconds. The exact separatrix takes infinite time to reach the saddle, so numerical crossings within the energy-error margin are unresolved.

A separatrix separates confined and crossing orbits

BackgroundA 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 mattersThe 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.

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

\begin{aligned}U(x)&=B\left[\left(\frac{x}{a}\right)^2-1\right]^2\\ &\quad+fx\end{aligned}

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

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

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

\begin{aligned}m\ddot x&=-U\prime(x)-c\dot x,\\ H(t)+Q(t)&=H(0),\\ Q(t)&=\int_0^t c\dot x^2\,dt\end{aligned}

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

\begin{aligned}\Delta H&=H-U(x_s),\\ T(H)&=\sqrt{2m}\\ &\quad\times\int_{x_-}^{x_+}\frac{dx}{\sqrt{H-U(x)}}\end{aligned}

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

\begin{aligned}U(x)&\simeq U(x_s)\\ &\quad-\frac{m\lambda_s^2}{2}(x-x_s)^2,\\ \lambda_s^2&=-\frac{U\prime\prime(x_s)}{m}\end{aligned}

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 misconceptionEvery crossing recorded at the critical energy really happens.

Better mental modelThe 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 misconceptionThe symmetric barrier setting always equals the barrier height on the launch side.

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

Damping changes the orbit

Typical misconceptionAfter the particle loses energy, the measured period still equals the undamped reference period.

Better mental modelAs 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

01
Compare trapped and crossing motion
Predict whether each release can reach the other well. Choose Small well oscillations and Cross both wells, then compare the phase portraits and the full-record crossing counts in the checks.
What to observe: Without damping, only motion above the barrier reaches both wells.
02
Approach the separatrix
Compare Almost reach the saddle with Creep across the saddle. Set Launch energy / barrier scale to exactly one in the symmetric, undamped case released from a minimum, and inspect the numerical energy balance error before interpreting a crossing.
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.
03
Let damping capture the particle
Choose Lose energy and settle. Predict which way the phase orbit will move, then open the checks and compare mechanical energy plus dissipated energy with the starting energy.
What to observe: Heat plus mechanical energy stays equal to the starting energy, and the orbit ends up trapped on one side.
04
Tilt without erasing a well
Choose Unequal wells, then drag the release point in phase space. Compare the actual saddle energy and release-side barrier with the symmetric scale in the checks.
What to observe: The saddle shifts and the barrier is higher from one side than from the other, yet both minima remain.