Simple harmonic motion as an ODE

Simple harmonic motion occurs when displacement from a stable equilibrium produces a restoring force proportional to the negative displacement. The sinusoidal motion is the solution of a differential equation, not a path that must be drawn in advance.

Linear restoring force

Choose \(x=0\) at the equilibrium of an ideal spring. Hooke's law is

\[F_x=-kx\]

with positive stiffness \(k\). If displacement is positive, force is negative; if displacement is negative, force is positive. The force always points toward equilibrium within the linear regime.

Newton's second law gives

\[m\ddot x+kx=0\]

or

\[\ddot x=-\omega_0^2x\qquad\omega_0=\sqrt{\frac{k}{m}}\]

Angular frequency has units radians per second. Ordinary frequency and period are

\[f=\frac{\omega_0}{2\pi}\qquad T=\frac{2\pi}{\omega_0}=2\pi\sqrt{\frac{m}{k}}\]

Increasing stiffness raises frequency; increasing mass lowers it. In this ideal linear model, amplitude does not affect frequency.

The analytic solution

The general real solution may be written

\[x(t)=C\cos(\omega_0t)+D\sin(\omega_0t)\]

For initial position \(x_0\) and velocity \(v_0\),

\[x(t)=x_0\cos(\omega_0t)+\frac{v_0}{\omega_0}\sin(\omega_0t)\]

and differentiation gives

\[v(t)=-\omega_0x_0\sin(\omega_0t)+v_0\cos(\omega_0t)\]

This pair is an exact reference for checking numerical output. It handles release from rest, an equilibrium crossing, or any combination of initial position and velocity.

An equivalent amplitude-phase form is

\[x(t)=A\cos(\omega_0t+\phi)\]

where

\[A=\sqrt{x_0^2+\left(\frac{v_0}{\omega_0}\right)^2}\]

and phase must be chosen with a quadrant-aware relation matching both initial values. Using a one-argument inverse tangent can select the wrong quadrant.

Energy exchange

Spring potential energy and kinetic energy are

\[U=\frac12kx^2\qquad K=\frac12mv^2\]

Their sum is constant:

\[E=\frac12mv^2+\frac12kx^2=\frac12kA^2\]

At turning points, speed is zero and energy is all potential. At equilibrium, potential is zero and speed magnitude is greatest:

\[v_{\max}=\omega_0A\]

Energy conservation assumes an isolated ideal oscillator with no damping, drive, friction, or time-dependent stiffness.

Turn the second-order equation into state updates

A numerical ODE solver commonly represents the second-order equation as two first-order equations:

\[\dot x=v\qquad\dot v=-\frac{k}{m}x\]

import numpy as np

mass, stiffness = 1.0, 4.0


def derivative(state):
    x, v = state
    return np.array([v, -stiffness * x / mass])


print(derivative([1.0, 0.0]))

The state at one time is the pair of position and velocity. Acceleration is derived from position. A precomputed sine curve may reproduce the ideal analytic answer but cannot be extended naturally to changing forces, collisions, or nonlinear behavior.

One simple numerical method

The kick-then-drift form of symplectic Euler updates velocity before position:

x, v, dt = 1.0, 0.0, 0.01
v -= stiffness * x / mass * dt
x += v * dt
print(x, v)

For this one undamped oscillator, its linear stability condition is

\[\omega_0\Delta t<2\]

The condition prevents unbounded numerical growth for this method and problem, but does not guarantee accurate phase or amplitude. Useful accuracy normally requires a much smaller fraction of the period.

Forward Euler updates both values from the old state. For the undamped oscillator it increases energy for every positive step and is unstable over repeated updates. A smooth short animation can hide that systematic error.

Velocity Verlet, Runge-Kutta methods, and implicit methods have different cost, phase, energy, and stability behavior. Name the method rather than saying only “Euler” or “numerical integration.”

Compare with the analytic reference

Generate analytic and numerical values at the same physical output times. Position error is

\[e_x(t)=x_{\mathrm{num}}(t)-x_{\mathrm{exact}}(t)\]

and relative energy error can be measured by

\[e_E(t)=\frac{E_{\mathrm{num}}(t)-E(0)}{E(0)}\]

when initial energy is nonzero. A method can keep amplitude bounded yet accumulate phase error, so compare zero crossings or fitted period as well as energy.

x, v, dt = 1.0, 0.0, 0.001
for step in range(2001):
    if step % 500 == 0:
        exact = np.cos(2 * step * dt)
        print(step * dt, x - exact, 0.5 * v * v + 2 * x * x)
    v -= 4 * x * dt
    x += v * dt

Repeat with half the time step and compare errors at identical output times. The observed error reduction should be consistent with the integrator's order once the asymptotic regime is reached.

Phase-space geometry

Eliminating time from the energy relation gives

\[\frac{x^2}{A^2}+\frac{v^2}{\omega_0^2A^2}=1\]

Thus a plot of velocity against position traces a closed ellipse. A numerical outward spiral signals energy gain; an inward spiral signals energy loss. A distorted but bounded loop can reveal phase or discretization error. The plot is diagnostic only when axes, units, and sampling times are preserved.

Simple harmonic motion beyond springs

Near a stable equilibrium of a smooth potential,

\[U(x)\approx U(x_0)+\frac12U''(x_0)(x-x_0)^2\]

so small motion is approximately harmonic with effective stiffness \(U''(x_0)\).

For a simple pendulum,

\[\ddot\theta+\frac{g}{L}\sin\theta=0\]

becomes harmonic only when \(\sin\theta\approx\theta\):

\[\omega_0\approx\sqrt{\frac{g}{L}}\]

At larger amplitude the period depends on amplitude, so the harmonic approximation must be checked rather than assumed.

Reliability checks

At positive displacement and zero velocity, acceleration must be negative. At zero stiffness, restoring acceleration vanishes. Doubling stiffness at fixed mass raises angular frequency by square root of two, while changing initial amplitude does not change ideal linear frequency.

Verify units, analytic initial conditions, energy exchange, phase-space shape, and time-step convergence. Keep numerical state unrounded and separate from rendering. Once the ideal oscillator passes these checks, damping, driving, and nonlinear force can be added one at a time with their own energy accounting.

Reading position, velocity, and acceleration together

Position, velocity, and acceleration are shifted in phase. At a positive turning point, position is largest, velocity is zero, and acceleration has its largest negative value. One quarter-period later the oscillator crosses equilibrium moving in the negative direction: position and acceleration are zero while speed is largest. At the negative turning point, velocity is again zero and acceleration is maximally positive.

These relationships are more reliable than memorizing three disconnected sine curves. They follow directly from the restoring law and derivatives. A stored numerical trace should reproduce the sign pattern in every quadrant of phase space. If velocity and position are both reported at different time levels by an update algorithm, the apparent phase relation can be misleading, so diagnostics should state exactly when each value is defined.

An equilibrium crossing is also a useful energy fixture. If the system starts from rest at amplitude \(A\), its first equilibrium crossing occurs after one quarter-period and its velocity points toward the side opposite the release. The speed magnitude should be \(\omega_0A\). Comparing this event with the numerical crossing tests event interpolation, energy conversion, and sign conventions at once.

Choosing a useful time step

A stability boundary is only the outer edge of usable behavior. Suppose a step gives only a few samples per period. The computed motion may remain bounded but represent the period poorly and miss turning points between samples. Choose output and internal steps from an accuracy target, then demonstrate convergence rather than relying on a universal sample count.

Long runs amplify small frequency error into a large phase error. For that reason, a one-period comparison and a hundred-period comparison answer different questions. A method intended for real-time qualitative motion may accept more phase error than a frequency-measurement calculation, but both should state the tolerance.

Parameter uncertainty also affects prediction. Because frequency depends on the square root of stiffness divided by mass, small fractional parameter errors contribute roughly half as strongly to fractional frequency error. Numerical refinement cannot eliminate uncertainty in a measured spring constant or effective moving mass. Keep model-parameter sensitivity separate from discretization error.

A worked Python notebook

A one kilogram mass on a spring of stiffness 4 newtons per metre starts one metre from equilibrium. The question for this calculation is precise: how does displacement change when we change time, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

This ideal oscillator has no damping or drive. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 3.14159 s. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for time, expressed in s. The output is displacement, expressed in m. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = \cos{\left(2 q \right)}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 0.785398 s, the expression gives 6.12323e-17 m. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("cos(2*q)", locals={"q": q})
reference_input = 0.7853981633974483
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted displacement:", reference_output, "m")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 1 m. The midpoint, at 1.5708 s, gives -1 m. The upper endpoint gives 1 m. Read these together with the assumptions: they describe different choices of time within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 1.5707963267948966, 3.141592653589793])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} s -> {output_value:.6g} m")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows time; the vertical axis shows displacement. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 3.141592653589793, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Time (s)")
ax.set_ylabel("Displacement (m)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does displacement change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.785398 s is -2 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = - 2 \sin{\left(2 q \right)}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.8011061266653973
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 0.801106 s. The tangent estimate is -0.0314159 m, while the full expression gives -0.0314108 m. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 3.141592653589793
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of time inside the stated interval, predict whether displacement will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: This ideal oscillator has no damping or drive. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of m: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(3.141592653589793) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 3.141592653589793, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

• OpenStax, University Physics Volume 1, Section 15.1, "Simple Harmonic Motion".
• OpenStax, University Physics Volume 1, Section 15.2, "Energy in Simple Harmonic Motion".
• MIT OpenCourseWare, 8.01SC Classical Mechanics, Chapter 23, Sections 23.1-23.3.
• Benjamin Crowell, Light and Matter, oscillation and spring discussions.
• The Physics Hypertextbook, "Simple Harmonic Motion".
• Physics LibreTexts, "Simple Harmonic Motion".