A Byte of PhysicsLogo

Conservative vs non-conservative forces

A conservative force stores and returns energy through a potential-energy function. Its work between two configurations depends only on the endpoints, not the route. A non-conservative interaction cannot generally be represented by one position-only potential over the relevant domain. The distinction concerns a force model and domain, not whether energy itself is globally conserved.

Equivalent tests under suitable conditions

For a time-independent force field on an appropriate domain, the following statements are equivalent:

The equations are

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

and

\[W_{A\to B}=\int_A^B\mathbf F\cdot d\mathbf r=U(A)-U(B)=-\Delta U\]

For a closed path,

\[\oint\mathbf F\cdot d\mathbf r=0\]

These tests need a stated domain. A curl-free field on a simply connected region is conservative under standard smoothness assumptions. If the domain excludes a line or point and contains a hole, local curl alone may not prove global path independence.

Potential differences matter, not the zero

Adding a constant to potential energy does not change force:

\[-\nabla(U+C)=-\nabla U\]

Choose a convenient zero and state it. Two references can report different potential values while agreeing on every potential difference and force. Comparing raw values without translating their zero conventions creates a false disagreement.

Uniform gravity

Near Earth's surface with upward coordinate \(y\),

\[U_g(y)=mgy+C\]

so

\[F_y=-\frac{dU_g}{dy}=-mg\]

Gravity's work from initial height \(y_i\) to final height \(y_f\) is

\[W_g=mg(y_i-y_f)\]

It does not depend on horizontal detours in this uniform-field model. The formula \(mgy\) belongs to a system including the body and the source of gravity; its simple form is not valid over arbitrary planetary distances.

Ideal spring

For displacement \(x\) from equilibrium,

\[U_s(x)=\frac12kx^2+C\]

and

\[F_s=-\frac{dU_s}{dx}=-kx\]

The square makes potential increase for compression or extension, while the derivative makes force point back toward equilibrium. The ideal model requires \(k>0\) and an elastic range where force is approximately linear.

Inverse-square gravity

For two point masses separated by \(r>0\), choose zero potential at infinite separation:

\[U(r)=-\frac{GMm}{r}\]

The negative value reflects the chosen zero and attraction. Radial force is

\[F_r=-\frac{dU}{dr}=-\frac{GMm}{r^2}\]

The model is singular at \(r=0\). Cancelling or differentiating symbolically must preserve this excluded point.

Mechanical energy conservation

If only conservative internal interactions do work and the potential has no explicit time dependence, mechanical energy

\[E_{\mathrm{mech}}=K+U\]

is constant:

\[\Delta K+\Delta U=0\]

This is a conditional statement. External work, friction, drag, driving, deformation, or time-dependent parameters can change mechanical energy. Total energy of a suitably closed larger system is still conserved; mechanical energy may become thermal or internal energy.

Friction and energy bookkeeping

Kinetic friction commonly does negative work on a sliding object's center-of-mass motion. For constant magnitude on a fixed surface over path length \(s\),

\[W_f=-\mu_kNs\]

This work depends on path length, not only endpoints. A round trip can return to the same position while friction has removed mechanical energy twice along the traversed distance.

The energy has not vanished. For a larger closed system containing both contacting surfaces, internal thermal energy rises. A useful ledger is

\[\Delta K+\Delta U+\Delta E_{\mathrm{thermal}}=W_{\mathrm{external}}\]

The exact terms depend on the system boundary. Calling friction “non-conservative” does not authorize discarding its energy transfer.

Drag and velocity dependence

A drag force such as

\[\mathbf F_D=-b\mathbf v_{\mathrm{rel}}\]

depends on velocity relative to a fluid and cannot be obtained from a position-only potential. Its instantaneous power on the body is

\[P_D=\mathbf F_D\cdot\mathbf v\]

For a stationary fluid in the chosen frame, this is \(-bv^2\le0\). With moving fluid, the sign for the body's kinetic energy can differ because the fluid can transfer energy to the body. The frame and larger system matter.

Time-dependent potentials

If potential depends explicitly on time, \(U(\mathbf r,t)\), force may still equal \(-\nabla U\), but mechanical energy changes as

\[\frac{d}{dt}(K+U)=\frac{\partial U}{\partial t}\]

Thus “force comes from a potential” does not alone guarantee constant mechanical energy. A changing spring stiffness or moving trap can inject or remove energy.

Numerical path-independence test

To test a candidate static force field, choose several paths between identical endpoints and numerically integrate work along each.

import numpy as np


def work_along(path, field):
    total = 0.0
    for start, end in zip(path[:-1], path[1:]):
        total += np.dot(field((start + end) / 2), end - start)
    return total


def spring_field(point):
    return -2.0 * point


straight = np.array([[0.0, 0.0], [1.0, 1.0]])
detour = np.array([[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]])
print(work_along(straight, spring_field))

first_work = work_along(straight, spring_field)
second_work = work_along(detour, spring_field)
assert np.isclose(first_work, second_work)
print("Same-endpoint work (J):", first_work, second_work)

A finite discrepancy can be quadrature error. Refine until each path result stabilizes before interpreting the difference. Agreement for two paths is evidence, not a proof for every possible path.

Closed loops and curl

In three-dimensional Cartesian coordinates, a smooth conservative field satisfies

\[\nabla\times\mathbf F=\mathbf 0\]

Numerical curl can expose local non-conservative behavior, but differentiation amplifies noise. Closed-loop work integrates the field and may be more robust in some data sets. Both need domain awareness, especially around singularities and excluded regions.

Verify force and potential implementations

When code contains both a force and a potential, do not maintain unrelated formulas without a consistency test. In one dimension, compare force with a centered numerical derivative:

\[F(x)\approx-\frac{U(x+h)-U(x-h)}{2h}\] 
def potential(x):
    return x**2


position, step = 0.7, 1e-4
numerical_force = -(potential(position + step) - potential(position - step)) / (
    2 * step
)
analytic_force = -2 * position
assert np.isclose(numerical_force, analytic_force)
print("Force (N):", numerical_force)

Repeat for several positions and step sizes away from discontinuities and singularities. Too large a step creates truncation error; too small a step can suffer cancellation.

Potential reconstruction from force

In one dimension, choose a reference point \(x_0\) and value \(U(x_0)\). Then

\[U(x)=U(x_0)-\int_{x_0}^{x}F(s)\,ds\]

In multiple dimensions, this construction is independent of integration path only for a conservative field on the relevant domain. A fitted potential also inherits measurement noise and integration uncertainty.

Classification pitfalls

Normal force from a fixed smooth track can do zero work on a constrained particle because force is perpendicular to velocity, but that does not make every normal force a position-gradient force. A moving track can do work. Static friction can do positive, negative, or zero work on a selected subsystem even while enforcing no slip. Magnetic force does no work on a point charge because it is perpendicular to velocity, yet it is velocity-dependent and not classified by an ordinary scalar potential alone.

“Does zero work in this motion” and “is conservative on this domain” are different questions.

Checks for energy classification

Conservative forces make position a reusable energy coordinate. Non-conservative interactions require a broader transfer ledger, but they do not violate conservation of total energy when the complete closed system is accounted for.

A worked Python notebook

Compute work done by the conservative spring force F equals minus 2x. The question for this calculation is precise: how does spring work from x=0 change when we change endpoint position, 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 work by the spring; work supplied quasistatically by an external hand has the opposite sign. 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 -2 to 2 m. 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 endpoint position, expressed in m. The output is spring work from x=0, expressed in J. 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.

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

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 m, the expression gives -1 J. 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("-q**2", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted spring work from x=0:", reference_output, "J")

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 -4 J. The midpoint, at 0 m, gives 0 J. The upper endpoint gives -4 J. Read these together with the assumptions: they describe different choices of endpoint position 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([-2.0, 0.0, 2.0])
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 -> {output_value:.6g} J")
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 endpoint position; the vertical axis shows spring work from x=0. 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(-2.0, 2.0, 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("Endpoint position (m)")
ax.set_ylabel("Spring work from x=0 (J)")
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 spring work from x=0 change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 m 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.

\[f'(q) = - 2 q\]

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 = 1.02
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, "J")
print("Full expression:", full_prediction, "J")
print("Approximation difference:", full_prediction - linear_prediction, "J")

Here the changed input is 1.02 m. The tangent estimate is -1.04 J, while the full expression gives -1.0404 J. 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 endpoint position inside the stated interval, predict whether spring work from x=0 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 work by the spring; work supplied quasistatically by an external hand has the opposite sign. 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 J: 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(2.0) - evaluate(-2.0))
for points in [101, 201, 401]:
    grid = np.linspace(-2.0, 2.0, 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

Share to: