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Numerical vs symbolic thinking

Symbolic and numerical methods are two ways of reasoning about the same physical model. Symbolic work manipulates relationships expressed with variables. Numerical work evaluates a model for particular values, often by replacing continuous change with many finite updates. Neither method is automatically more scientific. The useful question is which method answers the physical question with understandable error.

A problem with an exact answer

Take an object released from rest near Earth's surface. Choose downward as positive, assume constant gravitational acceleration, and ignore air resistance. Its acceleration is

\[a(t)=g\]

with initial conditions \(d(0)=0\) and \(v(0)=0\). Integrating the acceleration once gives velocity,

\[v(t)=gt\]

and integrating again gives distance,

\[d(t)=\frac{1}{2}gt^2\]

These are symbolic results. They describe every allowed value of \(t\) at once. With

\(g=9.81\,\mathrm{m/s^2}\)

and \(t=2.00\,\mathrm{s}\), the exact model predicts

\[d=\frac{1}{2}(9.81\,\mathrm{m/s^2})(2.00\,\mathrm{s})^2=19.62\,\mathrm{m}\]

"Exact" here means exact for the stated idealized differential equation. The rounded value of gravity and the assumptions about the real fall still limit physical accuracy.

Replace continuous change with finite updates

A numerical method stores a state at discrete times. Let \(\Delta t\) be a fixed step. Forward Euler uses values at the beginning of each step:

\[d_{n+1}=d_n+v_n\Delta t,\qquad v_{n+1}=v_n+g\Delta t\]

The two right sides must be evaluated from the old state. Mutating velocity first and then using the new velocity for position is a different method.

distance_metres = 0.0
velocity_metres_per_second = 0.0
gravity = 9.81  # m/s^2; downward is positive
time_step_seconds = 0.1
for step in range(20):
    old_velocity = velocity_metres_per_second
    distance_metres += old_velocity * time_step_seconds
    velocity_metres_per_second += gravity * time_step_seconds
print(f"After 2 s: {distance_metres:.3f} m")

def forward_euler_fall(
    end_time_seconds: float,
    time_step_seconds: float,
    gravity_metres_per_second_squared: float = 9.81,
) -> float:
    distance_metres = 0.0
    velocity_metres_per_second = 0.0

    steps = round(end_time_seconds / time_step_seconds)

    for step in range(0, steps):
        old_velocity = velocity_metres_per_second
        distance_metres += old_velocity * time_step_seconds
        velocity_metres_per_second += (
            gravity_metres_per_second_squared * time_step_seconds
        )

    return distance_metres

This example requires the requested duration to be an integer number of steps. A general implementation should shorten the last step rather than silently overshooting the requested time.

The update order changes the answer

The compact pair

gravity = 9.81
position = 0.0
velocity = 0.0
time_step = 0.1
for step in range(20):
    velocity += gravity * time_step
    position += velocity * time_step
print(f"Velocity-first result: {position:.3f} m")

is not forward Euler for both variables. It is the semi-implicit, or symplectic, Euler ordering: update velocity first, then use the new velocity to update position. For this fall, forward Euler underestimates the distance and semi-implicit Euler overestimates it.

For a total duration \(T=N\Delta t\), the two discrete results can be summed exactly:

\[d_{\mathrm{forward}}=\frac{1}{2}gT^2-\frac{1}{2}gT\Delta t\] \[d_{\mathrm{semi}}=\frac{1}{2}gT^2+\frac{1}{2}gT\Delta t\]

The error magnitude is proportional to the step size. At

\(T=2.00\,\mathrm{s}\)

and \(\Delta t=0.50\,\mathrm{s}\), forward Euler gives \(14.715\,\mathrm{m}\) and semi-implicit Euler gives \(24.525\,\mathrm{m}\). The analytic result, \(19.62\,\mathrm{m}\), lies between them.

Reducing the step to \(0.02\,\mathrm{s}\) gives \(19.4238\,\mathrm{m}\) and \(19.8162\,\mathrm{m}\). Both errors are smaller by a factor of twenty-five, matching the step-size reduction.

A numerical method can also respect the exact structure

Constant acceleration is special because the change over a step is known exactly:

\[d_{n+1}=d_n+v_n\Delta t+\frac{1}{2}g(\Delta t)^2\]

Together with the usual velocity update, this reproduces the analytic constant- acceleration result apart from floating-point rounding. Calling every loop-based method a rough approximation would therefore also be wrong. Error depends on the method and the model, not merely on the presence of a loop.

When symbolic thinking helps

Symbolic reasoning can expose scaling, invariants, and invalid inputs without choosing numbers. From the falling-distance equation, distance grows as the square of time and linearly with gravity. Units reveal that the output is a length. Differentiation recovers the velocity and acceleration. The closed form is fast to evaluate and gives a reference for testing other methods.

Symbolic methods become difficult or impossible when forces are nonlinear, geometry is irregular, material properties vary in space, boundaries move, or many bodies interact. Even when a symbolic expression exists, evaluating it may be numerically unstable in some form. Algebra is not automatically safe merely because it is exact on paper.

When numerical thinking helps

Numerical methods can work directly with measured tables, complicated force laws, and large coupled systems. They can show a trajectory at selected output times and let the same code handle many initial conditions. Their flexibility comes with obligations:

The claim "a smaller step is better" needs care. In a stable regime, reducing the step usually reduces truncation error. Eventually roundoff, cancellation, or another error source can dominate. Some explicit methods are unstable for steps above a problem- dependent limit. Convergence must be observed, not assumed.

Use both modes together

The strongest workflow is hybrid. Simplify symbolically where possible, use the symbolic result to predict units and limiting behavior, compute numerically where the remaining problem requires it, and compare the two in a case they share. The equation provides understanding and a test oracle; the numerical method extends the range of questions. Their disagreement is useful evidence about assumptions, implementation, or resolution rather than something to hide.

Read a refinement table carefully

A refinement table should compare results at the same physical time while changing only the step size. Include the numerical value, its signed difference from the reference, the absolute error, and the number of steps. If halving the step approximately halves the error for forward Euler in a smooth problem, the observed behavior agrees with its first-order global accuracy. One pair of runs is suggestive; several successive refinements are stronger evidence.

Do not compare states at different end times. Floating-point division may make the requested duration an imperfect multiple of the step, so the loop must define how it handles the remainder. Also do not round each state before the next update. Rounding is for reported output; feeding rounded display values back into the solver adds another error source that the mathematical method did not specify.

If refinement stops improving the answer, investigate instead of continuing blindly. The reference might be inaccurate, the code may have a defect, the method may be unstable, or roundoff may have become important. For a real experiment, uncertainty in the measured inputs can also be larger than the numerical change. There is little value in driving integration error far below uncertainty that the model cannot remove.

This is why a numerical answer should include method and resolution. A decimal without that information is not reproducible evidence. Symbolic and numerical results become most trustworthy when each is used to interrogate the other.

A worked Python notebook

Keep time symbolic, then evaluate the same falling-body expression on a numerical grid. The question for this calculation is precise: how does fall distance change when we change elapsed 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 is the constant-acceleration solution; a different force law requires a different expression. 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 0.1 to 3 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 elapsed time, expressed in s. The output is fall distance, 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.

\[f(q) = 4.905 q^{2}\]

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 1 s, the expression gives 4.905 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("9.81*q**2/2", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted fall distance:", 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 0.04905 m. The midpoint, at 1.55 s, gives 11.7843 m. The upper endpoint gives 44.145 m. Read these together with the assumptions: they describe different choices of elapsed 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([0.1, 1.55, 3.0])
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 elapsed time; the vertical axis shows fall distance. 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(0.1, 3.0, 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("Elapsed time (s)")
ax.set_ylabel("Fall distance (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 fall distance change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 s is 9.81 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 9.81 q\]

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 = 1.0145
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 1.0145 s. The tangent estimate is 5.04725 m, while the full expression gives 5.04828 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 = 2.9
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 elapsed time inside the stated interval, predict whether fall distance 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 is the constant-acceleration solution; a different force law requires a different expression. 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(3.0) - evaluate(0.1))
for points in [101, 201, 401]:
    grid = np.linspace(0.1, 3.0, 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.

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