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Rigid body approximation

A rigid body is an ideal object whose internal distances never change. Real materials deform, but the approximation is useful when those deformations are small enough that they do not materially affect the motion or forces of interest. It compresses the state of many particles into center-of-mass motion and orientation.

The rigidity condition

For any two material points \(i\) and \(j\) in a rigid body,

\[|\mathbf x_i(t)-\mathbf x_j(t)|=\text{constant}\]

The constraint removes internal stretching, compression, bending, shear, and vibration. The body may translate and rotate without changing shape. A ring remains a ring and a distance marked on it remains constant in every orientation.

Exact rigidity would require information to propagate instantly through matter, which is not physically possible. The rigid-body model is therefore an approximation, not a fundamental material law. It works when elastic-wave travel times and deformations are unimportant on the scale and accuracy of the problem.

Pose reconstructs every point

Choose body-frame coordinates \(\mathbf X\) measured from the center of mass. A world point is

\[\mathbf x(t)=\mathbf R_{\mathrm{cm}}(t)+\mathbf Q(t)\mathbf X\]

Here \(\mathbf R_{\mathrm{cm}}\) is translation and \(\mathbf Q\) is orientation. A proper rotation matrix satisfies

\[\mathbf Q^{\mathsf T}\mathbf Q=\mathbf I,\qquad \det\mathbf Q=1\]

These conditions preserve lengths and handedness. Reflections also preserve distances but are not continuously reachable rigid-body orientations from the identity.

import numpy as np

angle = np.pi / 3
rotation = np.array(
    [
        [np.cos(angle), -np.sin(angle), 0],
        [np.sin(angle), np.cos(angle), 0],
        [0, 0, 1],
    ]
)
centre = np.array([2.0, 3.0, 0.0])


def world_point(body_point):
    return centre + rotation @ body_point


print(world_point(np.array([1.0, 0.0, 0.0])))

World vertices are derived output. Moving one vertex independently changes the body shape and violates the approximation unless the body-frame geometry itself is intentionally redefined.

Degrees of freedom

An unconstrained rigid body in three dimensions has six configuration degrees of freedom: three for translation and three for orientation. A rotation matrix contains nine numbers but six constraints; a unit quaternion contains four numbers and one normalization constraint, with opposite quaternions representing the same orientation.

In planar motion, the configuration reduces to two translation coordinates and one angle. A fixed-axis rotor may need only one angle. Constraints such as hinges, sliders, and rolling contact reduce the independent motion further, but their reaction forces still enter the dynamics.

Center-of-mass translation

External forces control center-of-mass motion:

\[M\mathbf A_{\mathrm{cm}}=\sum\mathbf F_{\mathrm{ext}}\]

Internal forces maintain the body's structure but cancel from total linear momentum under the ordinary closed-system assumptions. A force through the center can translate the body without torque about the center. An off-center force generally produces both translation and rotation.

Rotation and the inertia tensor

Torque about the center changes spin angular momentum:

\[\frac{d\mathbf L_{\mathrm{cm}}}{dt}=\boldsymbol\tau_{\mathrm{cm},\mathrm{ext}}\]

In body coordinates,

\[\mathbf L_b=\mathbf I_b\boldsymbol\omega_b\]

The body-frame inertia tensor \(\mathbf I_b\) is constant for a truly rigid body. Its world representation changes with orientation:

\[\mathbf I_w=\mathbf Q\mathbf I_b\mathbf Q^{\mathsf T}\]

Mixing a world angular-momentum vector with an untransformed body inertia tensor produces incorrect angular velocity. The scalar relation \(L=I\omega\) applies only to an appropriate principal-axis or fixed-axis case.

Kinetic energy

Rigid-body kinetic energy separates into translation and rotation about the center:

\[K=\frac12M V_{\mathrm{cm}}^2+\frac12\boldsymbol\omega^{\mathsf T}\mathbf I\boldsymbol\omega\]

The tensor and angular velocity must be expressed in the same frame. This formula makes energy accounting possible without summing every material particle, provided the rigid approximation is valid.

Force application and torque

For an external force at world point \(\mathbf x_a\), torque about the center is

\[\boldsymbol\tau_{\mathrm{cm}}=(\mathbf x_a-\mathbf R_{\mathrm{cm}})\times\mathbf F\]

All forces contribute to net translation; their lever arms determine rotation. Contact forces may be unknown reactions constrained by nonpenetration and friction. A rigid body does not eliminate contact modeling—it eliminates internal deformation coordinates.

application_points = np.array([[3.0, 3.0, 0.0], [2.0, 4.0, 0.0]])
forces = np.array([[0.0, 5.0, 0.0], [2.0, 0.0, 0.0]])
print("Net force (N):", forces.sum(axis=0))
print("Net torque (N m):", np.cross(application_points - centre, forces).sum(axis=0))

When the approximation is appropriate

Compare expected deformation with the length scale and required accuracy. A steel tool under gentle handling may be effectively rigid for its overall trajectory even though it has microscopic strain. A long bridge, tuning fork, soft tire, or high-speed impact may need elastic modes because deformation changes forces and energy substantially.

Compare timescales too. If loading changes much more slowly than elastic waves cross the object, a quasistatic rigid or elastic approximation may work. Sudden impact can excite waves and local deformation before the rest of the object responds.

Mass distribution must remain fixed relative to the body. Fuel sloshing, moving limbs, deploying appendages, or rotating internal machinery changes the inertia and can exchange momentum with the shell. These systems need multiple bodies or internal coordinates.

Contacts reveal the limit

Perfectly rigid contact between ideal hard bodies produces an impulse over zero time and formally unbounded contact force. Impulse-based models can predict before-and-after momentum without predicting peak stress. Soft-contact models introduce compliance and contact duration.

Small numerical interpenetration is a property of an approximate contact method, not physical compression unless a material law connects overlap to force. Position correction can alter energy or momentum if it is not included in the ledger.

Orientation representations

Planar rotation can use an unwrapped scalar angle. Three-dimensional orientation uses a rotation matrix or quaternion to avoid treating finite rotations as additive vectors. Angular velocity is an instantaneous vector, but adding it componentwise to Euler angles is not generally a valid orientation update.

Numerical quaternion updates should preserve unit norm. Rotation matrices should preserve orthogonality and determinant. Renormalization controls representation drift but does not prove that angular momentum or orientation is physically accurate.

Verification

Choose several body-frame point pairs and compare their distances before and after reconstruction. Translate and rotate the pose; every internal distance must remain fixed. Round-trip body points to world coordinates and back.

first, second = np.array([1.0, 0.0, 0.0]), np.array([0.0, 2.0, 0.0])
body_distance = np.linalg.norm(first - second)
world_distance = np.linalg.norm(world_point(first) - world_point(second))
assert np.isclose(body_distance, world_distance)
print("Rigid transformation preserves distance:", world_distance)

For dynamics, test pure force through the center, a pure couple, torque-free rotation, and an off-center force. Reduce the time step and track linear momentum, angular momentum, energy transfer, orientation constraints, and contact residuals separately.

Failure signals

The model is questionable when measured shape changes exceed tolerance, vibration affects forces, stress or fracture is the quantity of interest, thermal expansion matters, or fluid-like internal motion changes inertia. Improving the time integrator cannot repair a missing deformation degree of freedom.

State the validity range rather than saying the object is simply rigid. The same object may be rigid enough for a thrown trajectory but not for sound, crashworthiness, or precision vibration analysis.

Reliability checks

The rigid-body approximation is a deliberate compression of matter into pose, mass, and inertia. It is accurate when shape changes are irrelevant to the question and reliable when geometry, frame transformations, contacts, and conservation ledgers are tested explicitly.

A worked Python notebook

A body point one metre from the centre rotates while the centre remains at world x equals 2 metres. The question for this calculation is precise: how does world horizontal coordinate change when we change orientation angle, 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.

Rigid motion preserves distances between body points and excludes deformation. 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 rad. 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 orientation angle, expressed in rad. The output is world horizontal coordinate, 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.

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

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 1.0472 rad, the expression gives 2.5 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("2+cos(q)", locals={"q": q})
reference_input = 1.0471975511965979
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "rad")
print("Predicted world horizontal coordinate:", 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 3 m. The midpoint, at 1.5708 rad, gives 2 m. The upper endpoint gives 1 m. Read these together with the assumptions: they describe different choices of orientation angle 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} rad -> {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 orientation angle; the vertical axis shows world horizontal coordinate. 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("Orientation angle (rad)")
ax.set_ylabel("World horizontal coordinate (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 world horizontal coordinate change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1.0472 rad is -0.866025 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.

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

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 = 1.0629055144645467
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 1.06291 rad. The tangent estimate is 2.4864 m, while the full expression gives 2.48634 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 orientation angle inside the stated interval, predict whether world horizontal coordinate 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: Rigid motion preserves distances between body points and excludes deformation. 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

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