Distinguish allowed positions from a finite-time crossing
The unit-mass conservative model U=x⁴/4−x² has two minima at±sqrt2 with energy−1 J and a central barrier U = 0=0. Mechanical energy gives K=E−U and speed magnitude sqrt(2(E−U)). Any position with U>E is forbidden in this model, but an allowed-position set need not be one connected interval.
Learning outcome: Solve turning roots for below-barrier, exact-barrier and above-barrier energies; retain disconnected intervals and the selected initial well. Explain the separate timing limitation.
Starting recap
Use vector dot products for work, gradients for conservative force, definite integrals for accumulation and explicit energy/frame boundaries. Check accessible states before assigning trajectories.
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At E=.25, the outer turning magnitude is sqrt(2+sqrt5)≈2.058171 m and the center is allowed with speed sqrt(.5) m/s. Compare with E = 0, whose barrier speed vanishes asymptotically.
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At E = 0, starting away from the barrier in this ideal double well, central crossing:
Experiment
Solve turning roots for below-barrier, exact-barrier and above-barrier energies; retain disconnected intervals and the selected initial well. Explain the separate timing limitation.
Keep disconnected allowed energy intervals separate
Worked teaching model · synthetic values · Unit-mass particle in U=x⁴/4−x² J, initially at selected minimum x=±sqrt2 with compatible nonnegative speed magnitude. Energy minimum−1 J and central barrier0 J. This is an accessibility diagram, not an integrated trajectory; E0 approaches the barrier asymptotically and does not cross it in finite time.
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Allowed positions require total energy at least potential
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: double-well potential (solid).2: selected total energy (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Allowed positions require total energy at least potential
double-well potential
| position (m) | energy (J) |
|---|---|
| -2.5 | 3.515625 |
| -2.416667 | 2.686933 |
| -2.333333 | 1.966049 |
| -2.25 | 1.344727 |
| -2.166667 | 0.8150077 |
| -2.083333 | 0.369225 |
| -2 | 0 |
| -1.916667 | -0.2997565 |
| -1.833333 | -0.5368441 |
| -1.75 | -0.7177734 |
| -1.666667 | -0.8487654 |
| -1.583333 | -0.9357518 |
| -1.5 | -0.984375 |
| -1.416667 | -0.9999879 |
| -1.333333 | -0.9876543 |
| -1.25 | -0.9521484 |
| -1.166667 | -0.8979552 |
| -1.083333 | -0.8292704 |
| -1 | -0.75 |
| -0.9166667 | -0.6637611 |
| -0.8333333 | -0.5738812 |
| -0.75 | -0.4833984 |
| -0.6666667 | -0.3950617 |
| -0.5833333 | -0.3113305 |
| -0.5 | -0.234375 |
| -0.4166667 | -0.1660759 |
| -0.3333333 | -0.1080247 |
| -0.25 | -0.06152344 |
| -0.1666667 | -0.02758488 |
| -0.08333333 | -0.006932388 |
| 0 | 0 |
| 0.08333333 | -0.006932388 |
| 0.1666667 | -0.02758488 |
| 0.25 | -0.06152344 |
| 0.3333333 | -0.1080247 |
| 0.4166667 | -0.1660759 |
| 0.5 | -0.234375 |
| 0.5833333 | -0.3113305 |
| 0.6666667 | -0.3950617 |
| 0.75 | -0.4833984 |
| 0.8333333 | -0.5738812 |
| 0.9166667 | -0.6637611 |
| 1 | -0.75 |
| 1.083333 | -0.8292704 |
| 1.166667 | -0.8979552 |
| 1.25 | -0.9521484 |
| 1.333333 | -0.9876543 |
| 1.416667 | -0.9999879 |
| 1.5 | -0.984375 |
| 1.583333 | -0.9357518 |
| 1.666667 | -0.8487654 |
| 1.75 | -0.7177734 |
| 1.833333 | -0.5368441 |
| 1.916667 | -0.2997565 |
| 2 | 0 |
| 2.083333 | 0.369225 |
| 2.166667 | 0.8150077 |
| 2.25 | 1.344727 |
| 2.333333 | 1.966049 |
| 2.416667 | 2.686933 |
| 2.5 | 3.515625 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
selected total energy
| position (m) | energy (J) |
|---|---|
| -2.5 | -0.5 |
| 2.5 | -0.5 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Initial minimum position | 1.414214 | m |
| Initial speed magnitude | 1 | m/s |
| Outer turning magnitude | 1.847759 | m |
| Inner turning magnitude for separated wells | 0.7653669 | m |
| Finite-time crossing of central barrier energetically possible | 0 | flag |
| Total energy above barrier | -0.5 | J |
The allowed set has separate left and right intervals; the selected initial well cannot cross the central barrier.
Solving U=E via z=x² gives z=2±2sqrt(1+E). Negative energies have two positive roots and two disconnected position intervals; at E−1 each interval shrinks to a minimum. At E0, speed near the barrier is proportional to |x|, so integrating dt=dx/v diverges logarithmically. Energy accessibility alone gives neither direction nor a finite travel time, and no trajectory is numerically integrated here.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The allowed set has separate left and right intervals; the selected initial well cannot cross the central barrier.
Solving U=E via z=x² gives z=2±2sqrt(1+E). Negative energies have two positive roots and two disconnected position intervals; at E−1 each interval shrinks to a minimum. At E0, speed near the barrier is proportional to |x|, so integrating dt=dx/v diverges logarithmically. Energy accessibility alone gives neither direction nor a finite travel time, and no trajectory is numerically integrated here.
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