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Force models as functions

A force model maps a physical state and parameters to a vector force. Writing that map as a small function makes assumptions, units, reference frames, and singularities easier to inspect. It also keeps the interaction law separate from the numerical method that advances motion.

A force function contract

For a body state \(\mathbf y=(\mathbf r,\mathbf v)\), a general force may be written

\[\mathbf F=\mathbf F(\mathbf r,\mathbf v,t;\boldsymbol\alpha)\]

The parameter set

\(\boldsymbol\alpha\)

can include masses, charges, spring constants, fluid properties, or field data. A useful software contract states:

import numpy as np


def spring_force(position, stiffness=10.0):
    position = np.asarray(position, dtype=float)
    if stiffness < 0 or not np.all(np.isfinite(position)):
        raise ValueError("Use nonnegative stiffness and finite positions.")
    return -stiffness * position


print("Spring force (N):", spring_force([0.1, 0.0]))

Uniform gravitational force

Near Earth's surface over a modest height range, gravitational field can be approximated as constant:

\[\mathbf F_g=m\mathbf g\]

If vertical is positive upward, \(\mathbf g=(0,-g)\). The model needs body mass and a field vector but not velocity. Its force function should return force, not acceleration; the dynamics layer divides the summed force by mass.

mass_kg = 2.0
gravitational_field = np.array([0.0, -9.81])
weight_n = mass_kg * gravitational_field
print("Weight (N):", weight_n)

This approximation fails over planetary-scale distance or where the field varies materially. The failure belongs in model metadata rather than being hidden inside a constant named gravity with unclear units.

Central inverse-square gravity

For point masses \(m_1\) and \(m_2\) at distinct positions,

\[\mathbf F_{2\to1}=Gm_1m_2\frac{\mathbf r_2-\mathbf r_1}{|\mathbf r_2-\mathbf r_1|^3}\]

The displacement points from target to source, so the positive coefficient produces attraction. The formula is undefined at zero separation. Returning zero there would replace a singular model with a false no-force claim.

G = 6.67430e-11
source_position = np.array([0.0, 0.0])
target_position = np.array([6.371e6, 0.0])
source_mass, target_mass = 5.972e24, 2.0
displacement = source_position - target_position
distance = np.linalg.norm(displacement)
if distance == 0:
    raise ValueError("Point masses cannot overlap.")
force_n = G * source_mass * target_mass * displacement / distance**3
print("Gravitational force (N):", force_n)

If finite-size bodies are intended, replace the point model with an appropriate interior or contact model. Adding a softening length changes the law and must be reported.

Ideal spring force

For a one-dimensional ideal spring,

\[F_s=-k(x-x_{\mathrm{eq}})\]

The spring constant \(k>0\) has units of force per length. The minus sign makes the force restoring: positive displacement produces negative force and vice versa.

For a spring between two points in multiple dimensions, let \(\mathbf d=\mathbf r_B-\mathbf r_A\), length \(\ell=|\mathbf d|\), rest length \(\ell_0\), and direction \(\hat{\mathbf d}=\mathbf d/\ell\). The force on A is

\[\mathbf F_{B\to A}=k(\ell-\ell_0)\hat{\mathbf d}\]

The direction is undefined when endpoints coincide. A zero rest length does not remove the need for an explicit policy at zero separation. Real springs also have elastic limits, mass, damping, and geometry absent from Hooke's linear model.

Drag depends on relative velocity

Linear drag is often written

\[\mathbf F_D=-b(\mathbf v-\mathbf v_{\mathrm{fluid}})\]

Quadratic drag can be written

\[\mathbf F_D=-\frac12\rho C_DA|\mathbf v_{\mathrm{rel}}|\mathbf v_{\mathrm{rel}}\]

where \(\mathbf v_{\mathrm{rel}}=\mathbf v-\mathbf v_{\mathrm{fluid}}\). The force opposes motion through the fluid, not necessarily motion relative to ground. The quadratic coefficient depends on density, drag coefficient, and reference area. The drag coefficient can itself vary with Reynolds number, orientation, and flow regime.

body_velocity = np.array([3.0, 0.0])
fluid_velocity = np.array([1.0, 0.0])
density, drag_coefficient, area = 1.2, 0.5, 0.1
relative_velocity = body_velocity - fluid_velocity
drag_n = (
    -0.5
    * density
    * drag_coefficient
    * area
    * np.linalg.norm(relative_velocity)
    * relative_velocity
)
print("Drag force (N):", drag_n)
assert np.dot(drag_n, relative_velocity) <= 0

At zero relative speed this expression naturally returns zero without normalizing the zero vector.

Contact and friction are state-dependent modes

Normal contact is not generally a force defined everywhere in space. It is active when surfaces touch and the unconstrained motion would violate nonpenetration. Its magnitude is solved from dynamics and the contact constraint.

Static friction obeys an inequality

\[|f_s|\le\mu_sN\]

rather than a single magnitude formula. A friction function must know whether contact is sticking or sliding. It first computes the force required for sticking and compares it with the limit. If the limit is exceeded, the state changes to sliding and a kinetic law applies. This is a piecewise constraint problem, not a smooth function of velocity with a guessed sign at zero.

Superposition and force accumulation

When a model supports superposition, net force is the vector sum

\[\mathbf F_{\mathrm{net},i}=\sum_j\mathbf F_{j\to i}+\sum_k\mathbf F_{k,\mathrm{external}}\]

Evaluate every term from the same snapshot. Pair interactions should return equal and opposite forces to their two targets. Summing magnitudes and assigning one direction is wrong except in a one-dimensional case where signs already encode direction.

contributions = {"weight": weight_n, "drag": drag_n}
net_force = sum(contributions.values(), start=np.zeros(2))
for name, force in contributions.items():
    print(name, force, "N")
print("Net force (N):", net_force)

Keeping the audit trail permits a free-body diagram and allows unexpectedly large or canceling forces to be inspected.

Conservative-force check

When a force comes from a time-independent potential energy,

\[\mathbf F=-\nabla U\]

the force implementation can be compared with a symbolic or finite-difference gradient of the implemented potential. For one coordinate,

\[F(x)\approx-\frac{U(x+h)-U(x-h)}{2h}\]

Agreement over a useful range of step sizes checks sign and derivative consistency. It does not prove that the chosen potential models the real system.

Determinism, caching, and state

A force function should not mutate its input state. Otherwise the result can depend on the order in which bodies or forces are evaluated. Caching is safe only when the cache key includes every relevant position, velocity, time, parameter, and environmental state. A field that changes with time makes a position-only cache stale.

Random microscopic forcing requires an explicit stochastic model and controlled random stream. It should not appear as hidden nondeterminism in an otherwise ordinary force function.

Validation tests

A force function is trustworthy when its physical source, target, domain, units, and assumptions are as explicit as its returned numbers. The dynamics layer can then combine forces and test alternative integrators without rewriting the physics.

A worked Python notebook

Use a quadratic drag model along the positive direction of relative motion. The question for this calculation is precise: how does drag force change when we change speed relative to air, 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.

Density, area, and drag coefficient are fixed; the signed expression shown is for nonnegative relative speed. 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 10 m/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 speed relative to air, expressed in m/s. The output is drag force, expressed in N. 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.

\[f(q) = - 0.03 q^{2}\]

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 3 m/s, the expression gives -0.27 N. 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("-0.03*q**2", locals={"q": q})
reference_input = 3.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m/s")
print("Predicted drag force:", reference_output, "N")

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 N. The midpoint, at 5 m/s, gives -0.75 N. The upper endpoint gives -3 N. Read these together with the assumptions: they describe different choices of speed relative to air 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, 5.0, 10.0])
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} m/s -> {output_value:.6g} N")
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 speed relative to air; the vertical axis shows drag force. 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, 10.0, 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("Speed relative to air (m/s)")
ax.set_ylabel("Drag force (N)")
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 drag force change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 3 m/s is -0.18 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.

\[f'(q) = - 0.06 q\]

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 = 3.05
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, "N")
print("Full expression:", full_prediction, "N")
print("Approximation difference:", full_prediction - linear_prediction, "N")

Here the changed input is 3.05 m/s. The tangent estimate is -0.279 N, while the full expression gives -0.279075 N. 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 = 10.0
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 speed relative to air inside the stated interval, predict whether drag force 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: Density, area, and drag coefficient are fixed; the signed expression shown is for nonnegative relative speed. 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 N: 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(10.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 10.0, 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.

Sources

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