One-Dimensional Potential Landscape

Can you predict a particle's motion just from a graph of its potential energy? Most of it. Draw a line at the total energy: the particle can only be where the landscape lies below that line, and it turns around exactly where the two meet. The curvature at the bottom of a valley gives only the frequency of small oscillations. A valley as flat as a fourth-power curve has zero curvature there, so that frequency comes out zero, yet the valley still traps the particle.

Shape a potential, drag the particle and release it. Watch force and velocity change, then read its turning points, energy and phase-space orbit.

Particle position $0$

Velocity $0$

Mechanical energy $0$

Recorded oscillation period $0$

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 $0$

Where will the particle turn?

Quadratic stiffness

k

2\,\mathrm{N\,m^{-1}}

Quartic confinement

q

1\,\mathrm{N\,m^{-3}}

Potential tilt

f

0\,\mathrm{N}

Release velocity

v_0

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

Release position

x_0

1\,\mathrm{m}

Particle mass

m

1\,\mathrm{kg}

Record duration

t_{\mathrm{record}}

24\,\mathrm{s}

Requested time step

h

6\,\mathrm{ms}

Selected time

t

1.92\,\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 polynomial force along a horizontal constraint. The guide and sphere are schematic; rolling and an actuator are not modelled. The energy curve is a graph, not a physical track. Gray phase contours use release energy; teal curves are numerical trajectories. Each trial uses one length scale and separate fixed force and velocity scales. Energy plus dissipated heat is audited at every integration step; the half-step position comparison covers at most the first ten seconds.

The energy graph predicts a whole trajectory

BackgroundA potential-energy graph gives the energy stored at each position. Its slope gives the force, its flat points are equilibria, and a horizontal line at the total energy marks every position a classical particle can reach. The curve is not a track: the particle moves along the horizontal axis, and the vertical axis is energy.

Why it mattersOpenStax University Physics, section 8.4, uses potential-energy diagrams to find stable points and turning points. This Lab goes further: you can reshape the well, follow the motion in phase space, compute the period by integration and check how well the simulation conserves energy.

Focus question: Which predictions still hold when the bottom of the well is not shaped like a parabola?
One-sentence intuition: The turning points come from the full energy equation, but the small-oscillation frequency depends only on how sharply the well curves at its bottom.

Core mathematical model

A potential you can reshape

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

The quadratic and quartic coefficients are never negative. If at least one of them is positive, the well traps the particle. The tilt term moves the equilibrium to one side.

Force and stability

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

If the curvature at the minimum is positive, small oscillations are simple harmonic. If the curvature is zero, the higher-order terms decide whether the minimum is stable.

Allowed positions and turning points

\begin{aligned}E&=\frac12mv^2+U(x),\\ U(x)&\le E,\\ U(x_\pm)&=E\end{aligned}

The speed is zero at a turning point. A particle released at rest at the equilibrium has an allowed region that shrinks to a single point.

Period at finite amplitude

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

The Lab evaluates this integral separately from the simulation. A cosine substitution removes the simple blow-up at each turning point, and the readout shows how much the result changes at half the resolution.

Energy drift and the small-oscillation reference

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

The reference curve holds only near the equilibrium. When the small-oscillation frequency is zero, the reference is motion at constant velocity; that is the linear approximation, and it does not say the real minimum is neutrally stable.

Common difficulties

Energy is not height

Typical misconceptionThe particle slides downhill along the plotted curve.

Better mental modelThe vertical axis is energy. The particle actually moves back and forth along the horizontal axis.

Force comes from the slope

Typical misconceptionThe force is largest where the potential energy is highest.

Better mental modelThe force depends on the slope at that point, not on the height. In a tilted well the equilibrium can even sit at a nonzero potential energy.

Zero curvature calls for higher-order terms

Typical misconceptionIf the small-oscillation frequency is zero, the minimum must be unstable.

Better mental modelA positive quartic term still traps the particle. It oscillates with a finite period that depends on the amplitude.

Run the experiment

01
Check a harmonic well
Predict the harmonic period, choose Harmonic motion, then open Period, force and numerical checks to compare the two period readouts.
What to observe: The period computed between the turning points matches the textbook harmonic period and the time between repeated turns in the trajectory.
02
Reshape the well
Predict which way the force points at the release point. Change Quadratic stiffness and Quartic confinement, then compare the force with the local linear prediction in the collapsed checks.
What to observe: The release energy and the turning points change together, and the controls show the actual coefficient values.
03
Flatten the minimum
Choose Flat-bottomed well. Predict the effect of halving the release displacement, then drag the particle closer to the minimum.
What to observe: The small-oscillation frequency stays at zero, but the actual period gets longer as the release gets smaller.
04
Move the equilibrium
Choose Shift the minimum. Drag the dot in the phase plot horizontally and vertically, then compare the energy boundaries before and after.
What to observe: The equilibrium moves. Changing the starting velocity changes the allowed interval, but the potential itself stays the same.