Energy accounting in code

An energy ledger is one of the strongest ways to test a physical model. It asks whether energy remained in named stores, crossed a declared boundary, or changed through a modeled source or sink. A decreasing mechanical-energy graph is not automatically an error, and a constant total is not automatically proof that the equations are correct.

First choose the system and stored energy

The word system identifies which matter and fields are inside the accounting boundary. For a closed mechanical and thermodynamic system, stored energy might be

\[E_{\mathrm{system}}=U+K_{\mathrm{cm}}+E_{\mathrm{external\ potential}}+E_{\mathrm{field}}+\cdots\]

where \(U\) is distinct from bulk center-of-mass kinetic energy and any external potential energy assigned to the system. Only include terms that the model actually represents, but list important omissions.

Changing the boundary changes classification. Friction between a block and an external surface transfers energy out of a block-only system. For a system containing block and surface, the same interaction can redistribute energy internally. The physical event did not change; the ledger boundary did.

Pick one transfer sign convention

This lesson treats every transfer into the selected system as positive. Define \(T_j\) for each heat, work, boundary flux, source, thermostat, or radiation category. Over one interval,

\[E_{\mathrm{expected,next}}=E_{\mathrm{previous}}+\sum_jT_j\]

The accounting residual is

\[r=E_{\mathrm{actual,next}}-E_{\mathrm{expected,next}}\]

For the thermodynamic convention \(\Delta U=Q-W_{\mathrm{by}}\), heat into the system contributes \(+Q\) and work done by the system contributes \(-W_{\mathrm{by}}\). Representing both as signed transfers makes the general ledger uniform.

A structured per-step record

stores_before = {"internal": 100.0, "kinetic": 5.0}
transfers_in = {"heat": 10.0, "work_by_system": -3.0}
stores_after = {"internal": 107.0, "kinetic": 5.0}
expected = sum(stores_before.values()) + sum(transfers_in.values())
residual = sum(stores_after.values()) - expected
print("Energy residual (J):", residual)

Store category values even when zero. This distinguishes “measured zero” from “not implemented.” Include units, time start, time end, integrator, step size, boundary policy, and model revision so the result can be reproduced.

Integrate rates over the matching interval

Power and heat-transfer rate have units of energy per time. They must be integrated before being added to energy stores:

\[T_j=\int_{t_n}^{t_{n+1}}P_j(t)\,dt\]

Using power at the beginning of a long step times the entire step can make the ledger less accurate than the state integrator. For smooth sampled endpoint powers, a trapezoidal estimate is

\[T_j\approx\frac{\Delta t}{2}\left[P_j(t_n)+P_j(t_{n+1})\right]\]

Use a time rule consistent with the state update. In a staggered scheme, forces, velocities, and fields may live at different time levels. Interpolate intentionally rather than multiplying unrelated samples.

Work from forces and moving boundaries

For a particle acted on by an external force,

\[W_{\mathrm{on}}=\int\mathbf F_{\mathrm{external}}\cdot d\mathbf r\]

For a gas doing quasistatic pressure–volume work,

\[W_{\mathrm{by}}=\int p_{\mathrm{external}}\,dV\]

Constraint forces may do no work for ideal stationary constraints, but moving constraints can transfer energy. A fixed wall reflecting an elastic particle changes momentum without changing kinetic energy. A moving wall can change particle energy and must receive the opposite ledger entry if it is outside the system.

Calculate work from the actual numerical trajectory or boundary displacement. Using an analytic endpoint formula from a different path can make a residual appear even when the state update is internally consistent.

Dissipation and thermostats are transfers or stores

Viscous drag can remove resolved mechanical energy. If the environment is outside the system, record a negative transfer. If a thermal reservoir is inside, add the corresponding internal-energy increase. Calling the missing energy “numerical damping” without checking the force work discards useful physics.

For linear drag \(\mathbf F_d=-b\mathbf v\), instantaneous power on the object is

\[P_d=\mathbf F_d\cdot\mathbf v=-b|\mathbf v|^2\le0\]

A thermostat also exchanges energy. Velocity rescaling, stochastic kicks, or feedback forces must report the net change they cause. For stochastic algorithms, conservation may hold only statistically for a larger system including the reservoir; report ensemble behavior as well as individual-step transfers.

Open boundaries and fields

Particles, fluid, or waves leaving a computational domain carry energy. Their loss from stored domain energy is expected only when an outward boundary-flux entry accounts for it. For a continuum energy density \(u\) and flux \(\mathbf S\),

\[\frac{d}{dt}\int_Vu\,dV=-\oint_{\partial V}\mathbf S\cdot d\mathbf A+P_{\mathrm{sources}}\]

With outward area orientation, positive outward flux lowers energy inside. Discrete face fluxes should be shared consistently between neighboring cells so an internal transfer cancels when totals are summed.

An open flow that transports mass may carry internal, kinetic, potential, and flow energy. A closed-system formula cannot be copied into a control-volume calculation without those mass-flow terms.

Residual tolerances

Floating-point arithmetic, measurement uncertainty, and numerical truncation prevent most real ledgers from producing exact zero. Use a combined tolerance such as

\[|r|\le\varepsilon_{\mathrm{abs}}+\varepsilon_{\mathrm{rel}}\max\left(|E_{\mathrm{previous}}|,|E_{\mathrm{actual,next}}|,|T|\right)\]

An absolute-only tolerance fails when the problem is rescaled. A relative-only tolerance becomes unstable near zero. The tolerance should reflect expected numerical order and input uncertainty, not be enlarged until a failing run passes.

Track a normalized residual for comparison:

\[r_{\mathrm{scaled}}=\frac{r}{E_{\mathrm{scale}}}\]

and record how \(E_{\mathrm{scale}}\) was chosen.

Per-step error and long-term drift

A tiny residual of the same sign on every update accumulates. Track both

\[R_N=\sum_{n=0}^{N-1}r_n\]

and the distribution of per-step values. Random roundoff-like errors may partially cancel; systematic bias produces drift. Plotting only the latest residual can hide this difference.

Energy behavior also depends on integrator type. Symplectic methods often keep energy error bounded for suitable conservative Hamiltonian systems without conserving the exact energy at each step. A method that monotonically improves one-step energy may still distort phase or other invariants. The ledger diagnoses behavior; it does not choose the correct method by itself.

Conservation can be wrong in a coordinated way

Code can conserve an incorrect quantity exactly. A force law and potential-energy formula with matching but incorrect coefficients may produce a perfect residual while predicting the wrong motion. A projection that renormalizes energy after each step can force the ledger to close while violating dynamics.

Therefore combine accounting with independent analytic cases, dimensional analysis, symmetry, convergence, and measured data. Never modify physical state solely to make a diagnostic look good unless that correction is itself a justified numerical method and is reported.

Verification fixtures

For a closed constant-velocity particle, kinetic energy should remain constant. For a particle in a uniform external force, compare work with the analytic kinetic-energy change. For an ideal harmonic oscillator, monitor kinetic plus potential energy across many periods and compare step-size refinements.

Add a known external energy transfer:

fixture = {
    "previous_energy": 100.0,
    "transfers": {"heater": 10.0, "work_by_system": -3.0},
    "actual_next_energy": 107.0,
}
expected = fixture["previous_energy"] + sum(fixture["transfers"].values())
assert expected == 107.0
assert fixture["actual_next_energy"] - expected == 0.0
print("Ledger fixture passed.")

Then omit the work entry and require the residual to expose exactly

\(-3\,\mathrm J\)

under the declared signs. Scale every energy by a common factor and verify the scaled tolerance behaves consistently.

For an open wave domain, accumulated outward flux should match the decrease in domain energy. For a damped oscillator, integrated drag power should account for lost mechanical energy. Reverse a system-boundary choice and confirm that classification changes while the combined larger-system balance remains consistent.

A worked Python notebook

Start with 100 joules, add heat, and account for 3 joules of work done by the system. The question for this calculation is precise: how does final stored energy change when we change heat supplied, 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.

The same system boundary and positive-inward transfer convention must be used for every term. 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 20 J. 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 heat supplied, expressed in J. The output is final stored energy, 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.

\[f(q) = q + 97\]

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 10 J, the expression gives 107 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("100+q-3", locals={"q": q})
reference_input = 10.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "J")
print("Predicted final stored energy:", 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 97 J. The midpoint, at 10 J, gives 107 J. The upper endpoint gives 117 J. Read these together with the assumptions: they describe different choices of heat supplied 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, 10.0, 20.0])
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} J -> {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 heat supplied; the vertical axis shows final stored energy. 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, 20.0, 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("Heat supplied (J)")
ax.set_ylabel("Final stored energy (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 final stored energy change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 10 J is 1 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.

\[f'(q) = 1\]

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 = 10.1
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 10.1 J. The tangent estimate is 107.1 J, while the full expression gives 107.1 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 = 20.0
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 heat supplied inside the stated interval, predict whether final stored energy 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: The same system boundary and positive-inward transfer convention must be used for every term. 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(20.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 20.0, 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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