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Curvilinear coordinates

Cartesian coordinates use fixed straight basis directions. Curvilinear coordinates use coordinate lines that can curve, and their local basis directions may change from point to point. Polar, cylindrical, and spherical coordinates often match the symmetry of a physical problem, but their motion formulas need more care than replacing variable names.

Polar coordinates in a plane

Polar coordinates describe a point by radial distance \(r\) from an origin and angle \(\phi\) measured from a chosen axis. With the conventional counterclockwise angle from positive horizontal,

\[x=r\cos\phi,\qquad y=r\sin\phi\]

and

\[r=\sqrt{x^2+y^2},\qquad \phi=\operatorname{atan2}(y,x)\]

The radial coordinate is normally nonnegative. Angles that differ by an integer number of complete turns represent the same direction. At the origin, \(r=0\), the angle is undefined because no radial direction is preferred.

import numpy as np

x, y = 3.0, 4.0
radius = np.hypot(x, y)
angle = np.arctan2(y, x) if radius > 1e-12 else None
print(f"Radius: {radius:.1f} m; angle: {angle:.3f} rad")
recovered = [radius * np.cos(angle), radius * np.sin(angle)]
assert np.allclose(recovered, [x, y])

The two array elements are not interchangeable physical components: radius has length units and angle is dimensionless in SI but carries an angular convention. Componentwise addition of polar coordinate pairs is not vector addition.

The local polar basis rotates with position

At each non-origin point, define a radial unit vector pointing outward and a transverse unit vector pointing toward increasing angle:

\[\hat{\mathbf e}_r=\cos\phi\,\hat{\mathbf i}+\sin\phi\,\hat{\mathbf j},\qquad \hat{\mathbf e}_\phi=-\sin\phi\,\hat{\mathbf i}+\cos\phi\,\hat{\mathbf j}\]

Unlike Cartesian unit vectors, these basis vectors depend on angle. Their time derivatives are

\[\frac{d\hat{\mathbf e}_r}{dt}=\dot\phi\,\hat{\mathbf e}_\phi,\qquad \frac{d\hat{\mathbf e}_\phi}{dt}=-\dot\phi\,\hat{\mathbf e}_r\]

These terms encode the fact that the local directions rotate as the object moves around the origin. Ignoring them would predict zero velocity for an object moving around a circle with constant radius, because its radial coordinate does not change. The changing basis supplies the tangential velocity.

Compare P and Q in Figure 1. At both points, the black radial arrow continues away from O, while the blue tangential arrow points counterclockwise. The pair turns as the point moves around the arc even though the Cartesian axes stay fixed. The drawing labels the angular direction with theta; it is the same direction denoted by phi in the polar equations above. The arrows illustrate orientation, not a change in the unit vectors' magnitudes.

Position and velocity in polar form

The position vector is

\[\mathbf r=r\hat{\mathbf e}_r\]

Differentiate both the scalar radius and its basis vector:

\[\mathbf v=\dot r\,\hat{\mathbf e}_r+r\dot\phi\,\hat{\mathbf e}_\phi\]

For motion from P toward Q along the circular arc in Figure 1, radius is constant, so the radial contribution is zero. Velocity still points along the local blue arrow: the angular contribution survives because the radial basis rotates. This is why differentiating the radius alone would miss the motion.

The radial component \(\dot r\) measures motion toward or away from the origin. The transverse component \(r\dot\phi\) has length per time units. Angular rate alone is not a linear velocity; points farther from the origin travel farther in the same angular interval.

Speed is

\[v=\sqrt{\dot r^{,2}+r^2\dot\phi^{,2}}\]

For uniform circular motion, \(r=R\) and \(\dot\phi=\omega\), so speed is \(R|\omega|\).

Acceleration in polar form

Differentiating velocity and again accounting for the moving basis gives

\[\mathbf a=(\ddot r-r\dot\phi^2)\hat{\mathbf e}_r+(r\ddot\phi+2\dot r\dot\phi)\hat{\mathbf e}_\phi\]

The radial term contains \(-r\dot\phi^2\), the inward centripetal acceleration. The transverse term includes angular acceleration \(r\ddot\phi\) and the coupling term \(2\dot r\dot\phi\). The latter appears because both radius and basis orientation change.

radius, radial_rate, radial_second_derivative = 2.0, 0.5, 0.0
angular_rate, angular_acceleration = 3.0, 0.0
velocity_polar = [radial_rate, radius * angular_rate]
acceleration_polar = [
    radial_second_derivative - radius * angular_rate**2,
    radius * angular_acceleration + 2 * radial_rate * angular_rate,
]
print("Radial/transverse velocity (m/s):", velocity_polar)
print("Radial/transverse acceleration (m/s^2):", acceleration_polar)

The returned pair is expressed in the local basis at that instant. To compare it with a Cartesian vector or with a polar vector at another location, convert both to a common basis.

Worked spiral example

Let radius grow at a constant rate while angle grows uniformly:

\[r(t)=r_0+ut,\qquad \phi(t)=\omega t\]

Then \(\dot r=u\), \(\ddot r=0\), \(\dot\phi=\omega\), and \(\ddot\phi=0\). Therefore

\[\mathbf v=u\hat{\mathbf e}_r+r\omega\hat{\mathbf e}_\phi\]

and

\[\mathbf a=-r\omega^2\hat{\mathbf e}_r+2u\omega\hat{\mathbf e}_\phi\]

Even though both coordinate rates are constant, acceleration is not zero. The local basis rotates, producing an inward term and a transverse coupling term. This example directly exposes why treating polar coordinates as fixed Cartesian array slots fails.

Cylindrical coordinates

Cylindrical coordinates extend plane polar coordinates with an axial coordinate:

\[x=r\cos\phi,\qquad y=r\sin\phi,\qquad z=z\]

Velocity is

\[\mathbf v=\dot r\hat{\mathbf e}_r+r\dot\phi\hat{\mathbf e}_\phi+\dot z\hat{\mathbf e}_z\]

and acceleration adds

\(\ddot z\hat{\mathbf e}_z\)

to the plane-polar result. This system is natural for motion around an axis, helical paths, pipes, and rotational symmetry.

Spherical coordinates and conventions

Spherical coordinates use one radius and two angles, but references do not universally agree on the angle names or which angle is measured from the positive vertical axis. One common physics convention uses polar angle \(\theta\) from positive vertical and azimuth \(\phi\) around that axis:

\[x=r\sin\theta\cos\phi,\quad y=r\sin\theta\sin\phi,\quad z=r\cos\theta\]

Some mathematics and engineering sources swap the symbols. Never infer convention from the words “spherical coordinates” alone. Store the angle definitions, ranges, handedness, and units with the data.

Spherical coordinates are singular at the origin, and azimuth is not unique on the axis. These are coordinate singularities: ordinary physical space need not be singular there. Algorithms must distinguish a coordinate ambiguity from a physical field singularity.

General scale factors

For orthogonal curvilinear coordinates \((q_1,q_2,q_3)\), a small displacement has the form

\[d\mathbf r=h_1dq_1\hat{\mathbf e}_1+h_2dq_2\hat{\mathbf e}_2+h_3dq_3\hat{\mathbf e}_3\]

The scale factors \(h_i\) convert coordinate changes to physical lengths. In plane polar coordinates, \(h_r=1\) and \(h_\phi=r\). That second scale factor is exactly why transverse speed is \(r\dot\phi\) rather than only angular rate.

Formulas for gradient, divergence, curl, volume, and numerical cell size depend on these scale factors. Reusing Cartesian finite differences on a spherical grid without metric terms generally gives wrong physical derivatives and conservation laws.

Angle wrapping and numerical continuity

A trajectory crossing the chosen angle branch cut can jump numerically from near \(\pi\) to near \(-\pi\) even though the physical direction changes smoothly. Differencing those wrapped values produces a false angular velocity near one full turn divided by the time step.

For continuous motion, unwrap the angle by choosing the equivalent angle closest to the previous sample before differentiating. Retain the wrapped angle for display if desired, but keep continuous internal phase separately.

wrapped_angles = np.deg2rad([170, 179, -179, -170])
unwrapped_angles = np.unwrap(wrapped_angles)
print("Continuous angles (degrees):", np.rad2deg(unwrapped_angles))

Checks for coordinate calculations

Curvilinear coordinates simplify geometry when they match a problem's symmetry. Their components are not fixed global directions, however. Correct kinematics follows from differentiating both the coordinates and the moving basis.

A worked Python notebook

A particle moves at fixed radius 2 metres while its angular speed changes between experiments. The question for this calculation is precise: how does radial acceleration change when we change angular speed, 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.

Each run is circular motion at fixed radius; the inward radial component is negative. 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 4 rad/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 angular speed, expressed in rad/s. The output is radial acceleration, expressed in m/s^2. 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.

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

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 rad/s, the expression gives -18 m/s^2. 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*q**2", locals={"q": q})
reference_input = 3.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "rad/s")
print("Predicted radial acceleration:", reference_output, "m/s^2")

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 m/s^2. The midpoint, at 2 rad/s, gives -8 m/s^2. The upper endpoint gives -32 m/s^2. Read these together with the assumptions: they describe different choices of angular speed 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, 2.0, 4.0])
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} rad/s -> {output_value:.6g} m/s^2")
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 angular speed; the vertical axis shows radial acceleration. 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, 4.0, 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("Angular speed (rad/s)")
ax.set_ylabel("Radial acceleration (m/s^2)")
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 radial acceleration change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 3 rad/s is -12 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.

\[f'(q) = - 4 q\]

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 = 3.02
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/s^2")
print("Full expression:", full_prediction, "m/s^2")
print("Approximation difference:", full_prediction - linear_prediction, "m/s^2")

Here the changed input is 3.02 rad/s. The tangent estimate is -18.24 m/s^2, while the full expression gives -18.2408 m/s^2. 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 = 4.0
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 angular speed inside the stated interval, predict whether radial acceleration 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: Each run is circular motion at fixed radius; the inward radial component is negative. 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/s^2: 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(4.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 4.0, 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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