M045 · Barrier / critical slowing

Double-Well, Barrier & Separatrix Lab

Tune the barrier, well spacing, asymmetry and damping. Compare confined oscillations with inter-well motion, approach the separatrix from both sides and capture an energetic particle through dissipation. Measure crossings and periods against independent energy quadrature.

Acquired time $0\,s$

Position $0\,m$

Velocity $0\,\mathrm{m\,s^{-1}}$

Conservative force $0\,N$

Potential energy $0\,J$

Kinetic energy $0\,J$

Mechanical energy $0\,J$

Dissipated energy $0\,J$

Current energy balance defect $0\,J$

Full-record maximum balance defect $0\,J$

Half-step displacement difference · first ten seconds $0\,m$

Actual integration step $0\,ms$

Nearest release-side minimum $0\,m$

Local undamped angular frequency $0\,\mathrm{rad\,s^{-1}}$

Allowed regions at release energy $1$

Release-region left boundary $0\,m$

Release-region right boundary $0\,m$

Conservative quadrature period $0\,s$

Last acquired full turn interval $\text{Await turns}$

Period quadrature resolution difference $0\,s$

Measured minus conservative period $0\,s$

Acquisition status $\text{Complete}$

Barrier above nearest release-side minimum $0\,J$

Saddle position $0\,m$

Current energy above saddle $0\,J$

Acquired numerical saddle crossings $0$

First acquired numerical crossing $\text{Not acquired}$

Release energy classification $\text{Above barrier}$

Acquired dissipative enclosure $\text{Unconfirmed}$

RELEASE / ACQUIRE / VERIFY

How close can you approach the barrier before the motion changes its topology?

Particle mass

m

1.00\,kg

Symmetric barrier scale

B

1.00\,J

Symmetric half separation

a

1.00\,m

Tilt relative to well-loss threshold

f/f_{\mathrm{crit}}

0.00\,

Viscous damping

c

0.00\,kg\,s^{-1}

Release position

x_0

-1.00\,m

Launch kinetic energy / barrier scale

K_0/B

1.08\,

Launch direction

\operatorname{sgn}v_0

1.00\,

Observation window

t_{\mathrm{obs}}

40.00\,s

Requested step

h

5.00\,ms

Acquired record

t/t_{\mathrm{obs}}

0.00\,\%

The critical preset has no finite conservative period. Numerical motion arbitrarily close to the saddle is sensitive to energy error; inspect the audit before calling a crossing physical. Exact saddle release remains at rest without a perturbation. The period chart always uses conservative motion from the nearest release-side minimum; damping changes the measured period. Enclosure is reported only after acquired energy falls below the saddle by more than numerical error.

All quantities use SI units for an ideal point particle in a prescribed polynomial potential. The vertical curve is an energy graph, not a physical track. Gray contours use release energy; acquired motion is teal. Playback reintegrates from saved checkpoints. Step refinement covers at most the first ten seconds; the energy audit covers every integration step.

Physics tutorial

A separatrix separates confined and crossing orbits

BackgroundA quartic double well has two stable minima and a saddle between them. Below the saddle energy, a conservative particle remains on one side. Above it, the allowed interval connects both wells. At critical energy, a nonstationary separatrix approaches the saddle asymptotically rather than completing a periodic orbit.

Why it mattersThe energy and phase-space construction follows classical one-coordinate mechanics. The barrier threshold and tilt limit here are derived directly from the displayed quartic potential. The viscous model books every removed kinetic-energy increment.

Start with the essentials

Focus question: Why does crossing a little above the barrier take so much longer?
One-sentence intuition: The particle spends an increasing time near the saddle as its energy approaches the barrier. Damping can remove enough energy to disconnect a previously crossing orbit.

Core mathematical model

Tilted double well

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

The symmetric barrier scale is measured from either minimum. With tilt, the displayed release-side barrier is recomputed from actual stationary points.

Persistence of two minima

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

The slider stays below the saddle-node threshold; the two minima and intervening saddle remain distinct.

Viscous energy ledger

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

Exponential viscous substeps account for their exact kinetic-energy loss. Conservative step error remains visible in the balance defect.

Barrier and conservative period

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

The critical orbit has no finite period. The plot leaves a gap instead of connecting the below- and above-barrier branches.

Local saddle dynamics

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}

Logarithmic period growth near the saddle amplifies small energy errors. Exact saddle release remains at rest without perturbation.

Common difficulties

Crossing is a numerical observation

Typical misconceptionEvery recorded crossing at critical energy is physical.

Better mental modelThe exact separatrix does not cross the saddle in finite time. Finite-step error can change its topology; compare energy and half-step defects.

A tilted scale is not the actual barrier

Typical misconceptionThe symmetric barrier parameter always equals the launch-side barrier height.

Better mental modelTilt changes the energies of both minima and the saddle; the readout uses their actual difference.

Damping changes the orbit

Typical misconceptionThe conservative reference period remains the measured period after energy loss.

Better mental modelDissipation changes the instantaneous orbit. Capture is certified only after acquired energy lies below the saddle beyond numerical uncertainty.

Run the experiment

01
Compare both topologies
Acquire trapped and crossing presets, then inspect phase space and acquired saddle events.
What to observe: Only above-barrier conservative motion connects the two wells.
02
Approach the boundary
Compare near-below and near-above presets, then select critical energy.
What to observe: Both periods grow; the critical reference reports no finite period and its numerical sensitivity grows sharply.
03
Capture by dissipation
Acquire the dissipative-capture preset and inspect heat plus mechanical energy.
What to observe: The energy ledger closes while the orbit eventually becomes enclosed on one side.
04
Tilt without erasing a well
Change tilt and drag the release in phase space; compare actual saddle energy with the symmetric scale.
What to observe: The saddle moves and the two launch-side barriers differ even though both minima persist.