M044 · Potential / turning points

One-Dimensional Potential Landscape

Edit curvature, quartic confinement and tilt. Drag a release point or set both position and velocity in phase space, then compare allowed regions, exact turning boundaries and finite-amplitude periods with a local harmonic prediction.

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 $\text{Complete}$

RELEASE / ACQUIRE / VERIFY

Can you predict a particle’s motion from the shape of its energy landscape?

Particle mass

m

1.00\,kg

Quadratic stiffness

k

2.00\,N\,m^{-1}

Quartic confinement

q

1.00\,N\,m^{-3}

Potential tilt

f

0.00\,N

Release position

x_0

1.00\,m

Release velocity

v_0

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

Observation window

t_{\mathrm{obs}}

24.00\,s

Requested step

h

6.00\,ms

Acquired record

t/t_{\mathrm{obs}}

0.00\,\%

Drag the purple release outline horizontally, or drag the phase-space outline in both directions. The orange square at minus one metre edits quadratic stiffness while other coefficients stay fixed. Every drag has keyboard sliders. A pure quartic minimum is stable but has zero linear frequency. If both restoring coefficients vanish, acquisition stops and retains the last valid plot.

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

The energy graph predicts a whole trajectory

BackgroundA prescribed potential assigns energy to position. Its slope gives force, its stationary points give equilibria, and a horizontal total-energy line bounds all classically allowed positions. The curve is not a shaped rail: the particle’s coordinate is horizontal, and the vertical coordinate is energy.

Why it mattersOpenStax University Physics section 8.4 connects potential diagrams to stability and turning points. This workbench adds editable confinement, a phase portrait, period quadrature and a numerical energy audit.

Start with the essentials

Focus question: Which predictions survive when a well stops being locally quadratic?
One-sentence intuition: Turning boundaries come from the full energy equation; a small-oscillation frequency comes only from local curvature.

Core mathematical model

Editable confining landscape

U(x)=\frac{kx^2}{2}+\frac{qx^4}{4}+fx

Nonnegative quadratic and quartic coefficients provide confinement when at least one is positive. The tilt shifts the equilibrium.

Force and stability

F(x)=-U\prime(x),\qquad U\prime(x_*)=0,\qquad \omega_0^2=\frac{U\prime\prime(x_*)}{m}

Positive curvature gives a local harmonic approximation. At a zero-curvature minimum, higher-order terms decide stability.

Allowed positions and turning boundaries

E=\frac12mv^2+U(x),\qquad U(x)\le E,\qquad U(x_\pm)=E

A turning boundary has zero speed. An equilibrium release can collapse the allowed region to a single point.

Finite-amplitude period

T(E)=\sqrt{2m}\int_{x_-}^{x_+}\frac{dx}{\sqrt{E-U(x)}}

Independent quadrature removes simple endpoint singularities with a cosine substitution. Its resolution difference is shown.

Energy audit and local reference

\delta E(t)=E(t)-E(0),\qquad x_{\mathrm{lin}}(t)=x_*+(x_0-x_*)\cos\omega_0t+\frac{v_0}{\omega_0}\sin\omega_0t

The reference is local. For zero linear frequency, the reference is free linear motion, not a claim of neutral nonlinear stability.

Common difficulties

Energy is not height

Typical misconceptionThe particle slides downhill along the plotted curve.

Better mental modelThe plot’s vertical coordinate is energy; actual motion evolves along the one-dimensional horizontal coordinate.

Force uses slope

Typical misconceptionThe largest potential value gives the largest force.

Better mental modelForce depends on the local derivative. A displaced equilibrium can occur at nonzero potential energy.

Zero curvature needs higher order

Typical misconceptionZero small-oscillation frequency proves instability.

Better mental modelA positive quartic term still confines the particle and yields finite amplitude-dependent periods.

Run the experiment

01
Benchmark a harmonic well
Select the harmonic preset and acquire a full record.
What to observe: The turning interval period agrees with the analytic harmonic period and the trajectory’s repeated turns.
02
Edit the apparatus
Drag the orange stiffness handle, change quartic confinement and compare force with its local linear prediction.
What to observe: The release energy and turning positions change together; controls expose the actual coefficients.
03
Flatten the minimum
Select the pure quartic preset, then reduce release displacement.
What to observe: The local linear frequency remains zero while the finite-amplitude period grows as the release shrinks.
04
Move the equilibrium
Select the tilted preset, then drag the phase-space release in both directions.
What to observe: The equilibrium shifts and velocity changes the allowed interval without changing the underlying potential.