Use the force derivative without discarding a flat minimum
A one-dimensional conservative potential defines force by F=−dU/dx. For U=−a x²/2+b x⁴/4 with positive b, force is a x−b x³. Equilibrium requires zero force, not zero potential. Additive potential constants never move these equilibria.
Learning outcome: Derive the force and stationary points for positive, zero and negative a. Check second derivatives and directly inspect the potential where that test vanishes.
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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For a = 1,b = 2, minima are at±1/sqrt2 m with potential−1/8 J and curvature2 N/m. For a−1,b = 2, only the origin remains and its curvature is+1 N/m.
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At a = 0,b>0, the origin of this exact quartic potential is:
Experiment
Derive the force and stationary points for positive, zero and negative a. Check second derivatives and directly inspect the potential where that test vanishes.
Derive equilibria from a nonlinear potential
Worked teaching model · synthetic values · One-dimensional conservative model U=−a x²/2+b x⁴/4 J, positive quartic b. Equilibria use F=−dU/dx; second-derivative test is inconclusive when curvature vanishes and the actual quartic must be inspected. No physical spring construction.
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Potential and its stationary-point structure
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: quartic potential (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Potential and its stationary-point structure
quartic potential
| position (m) | potential energy (J) |
|---|---|
| -2 | 0 |
| -1.92 | -0.2890138 |
| -1.84 | -0.5200282 |
| -1.76 | -0.6988186 |
| -1.68 | -0.8309146 |
| -1.6 | -0.9216 |
| -1.52 | -0.975913 |
| -1.44 | -0.9986458 |
| -1.36 | -0.994345 |
| -1.28 | -0.9673114 |
| -1.2 | -0.9216 |
| -1.12 | -0.8610202 |
| -1.04 | -0.7891354 |
| -0.96 | -0.7092634 |
| -0.88 | -0.6244762 |
| -0.8 | -0.5376 |
| -0.72 | -0.4512154 |
| -0.64 | -0.367657 |
| -0.56 | -0.2890138 |
| -0.48 | -0.217129 |
| -0.4 | -0.1536 |
| -0.32 | -0.09977856 |
| -0.24 | -0.05677056 |
| -0.16 | -0.02543616 |
| -0.08 | -0.00638976 |
| 0 | 0 |
| 0.08 | -0.00638976 |
| 0.16 | -0.02543616 |
| 0.24 | -0.05677056 |
| 0.32 | -0.09977856 |
| 0.4 | -0.1536 |
| 0.48 | -0.217129 |
| 0.56 | -0.2890138 |
| 0.64 | -0.367657 |
| 0.72 | -0.4512154 |
| 0.8 | -0.5376 |
| 0.88 | -0.6244762 |
| 0.96 | -0.7092634 |
| 1.04 | -0.7891354 |
| 1.12 | -0.8610202 |
| 1.2 | -0.9216 |
| 1.28 | -0.9673114 |
| 1.36 | -0.994345 |
| 1.44 | -0.9986458 |
| 1.52 | -0.975913 |
| 1.6 | -0.9216 |
| 1.68 | -0.8309146 |
| 1.76 | -0.6988186 |
| 1.84 | -0.5200282 |
| 1.92 | -0.2890138 |
| 2 | 0 |
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 |
|---|---|---|
| Origin force | 0 | N |
| Origin potential curvature | -2 | N/m |
| Origin is a strict potential minimum | 0 | flag |
| Positive nonzero equilibrium | 1.414214 | m |
| Negative nonzero equilibrium | -1.414214 | m |
| Potential at either minimum | -1 | J |
| Curvature at either nonzero minimum | 4 | N/m |
The origin is unstable and two nonzero minima are stable in the conservative local sense.
F=a x−b x³. Stationary points satisfy x(a−b x²)=0; curvature Usecond=−a+3b x². Positive curvature gives a local minimum, negative a maximum. At a0,b>0, U=b x⁴/4 rises on both sides despite zero second derivative, so treating zero curvature as automatically neutral would be wrong.
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The origin is unstable and two nonzero minima are stable in the conservative local sense.
F=a x−b x³. Stationary points satisfy x(a−b x²)=0; curvature Usecond=−a+3b x². Positive curvature gives a local minimum, negative a maximum. At a0,b>0, U=b x⁴/4 rises on both sides despite zero second derivative, so treating zero curvature as automatically neutral would be wrong.
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Why is local stability not the same as dissipative settling?
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