A Byte of PhysicsLogo

Energy-based motion solvers

In a one-dimensional conservative system, total energy can identify allowed positions, speed magnitudes, turning points, and travel times without directly stepping Newton's second law. This approach is powerful but incomplete: energy normally loses velocity direction, and in multiple dimensions it does not determine a unique trajectory.

Allowed kinetic-energy budget

For mass \(m>0\), total mechanical energy \(E\), and time-independent potential \(U(x)\),

\[E=\frac12mv^2+U(x)\]

so the kinetic-energy budget at a candidate position is

\[K(x)=E-U(x)\]

Classically allowed positions have \(K>0\). Turning points have \(K=0\). Positions with \(K<0\) are forbidden under the stated model and total energy.

def classify_position(energy, potential, tolerance=1e-12):
    budget = energy - potential
    if budget < -tolerance:
        return "forbidden", budget
    if abs(budget) <= tolerance:
        return "turning point", 0.0
    return "allowed", budget


print(classify_position(2.0, 0.5))

The tolerance has energy units and should reflect input and numerical uncertainty. A large negative budget must not be clamped away.

Speed has two direction branches

At an allowed position,

\[|v(x)|=\sqrt{\frac{2[E-U(x)]}{m}}\]

but velocity is

\[v(x)=\pm\sqrt{\frac{2[E-U(x)]}{m}}\]

Energy determines magnitude, not direction. A continuous trajectory stays on one branch until it reaches a turning point, where direction may reverse according to the force and local potential shape. Merely choosing a positive square root makes every trajectory move toward increasing coordinate.

import numpy as np


def velocity_from_energy(position, direction, energy=2.0, mass=1.0, stiffness=4.0):
    if direction not in [-1, 1]:
        raise ValueError("Choose direction -1 or +1.")
    status, budget = classify_position(energy, 0.5 * stiffness * position**2)
    if status == "forbidden":
        raise ValueError("Position lies outside the allowed interval.")
    return direction * np.sqrt(2 * budget / mass)


print(velocity_from_energy(0.5, 1), velocity_from_energy(0.5, -1))

Direction sign should be positive or negative and must be updated only by a physical turning or event rule.

Initial energy

Compute total energy from a complete initial state:

\[E=\frac12mv_0^2+U(x_0)\]

Potential and total must use the same additive zero. Shifting potential by a constant and shifting total energy by the same constant leaves \(E-U(x)\) unchanged. Changing only one creates false allowed or forbidden regions.

Turning points by root finding

Turning points solve

\[U(x)-E=0\]

Bracketed root finding is reliable when the function changes sign across a simple root. A root that merely touches the energy line can have even multiplicity and no sign change, so sampling extrema or using derivative information may be necessary.

import sympy as sp

x = sp.Symbol("x", real=True)
turning_points = sp.solve(sp.Eq(2 * x**2, 2), x)
print("Turning points (m):", turning_points)

After roots are found, test intervals between them to determine which connected regions are allowed. An initial state can explore only the allowed component connected to its starting position unless an external transfer changes energy.

Harmonic oscillator example

For \(U(x)=\tfrac12kx^2\), turning points satisfy

\[\frac12kx^2=E\]

so

\[x_{\mathrm{turn}}=\pm\sqrt{\frac{2E}{k}}\]

The amplitude is \(A=\sqrt{2E/k}\). Speed is largest at equilibrium, where potential is smallest, and vanishes at both endpoints. The period

\[T=2\pi\sqrt{\frac{m}{k}}\]

does not depend on energy in this quadratic potential. That amplitude independence is special; nonlinear potentials generally produce energy-dependent periods.

Travel time by quadrature

On a branch with positive velocity,

\[dt=\frac{dx}{v(x)}=\sqrt{\frac{m}{2}}\frac{dx}{\sqrt{E-U(x)}}\]

Therefore travel time from \(x_a\) to \(x_b\) on a monotonic branch is

\[t_b-t_a=\sqrt{\frac{m}{2}}\int_{x_a}^{x_b}\frac{dx}{\sqrt{E-U(x)}}\]

The integrand often diverges at an ordinary turning point because speed approaches zero, but the improper integral can remain finite. Evaluating exactly at the endpoint with a naive quadrature can produce infinity or NaN. A change of variable, analytic endpoint treatment, or quadrature designed for integrable singularities is safer.

# x = A sin(theta) removes the square-root singularity at the turning point.
mass, stiffness, amplitude = 1.0, 4.0, 1.0
omega = np.sqrt(stiffness / mass)
start, end = 0.0, 1.0
travel_time = (np.arcsin(end / amplitude) - np.arcsin(start / amplitude)) / omega
print(f"Time from centre to turning point: {travel_time:.6f} s")

Period from the potential

For bound motion between two ordinary turning points \(x_-\) and \(x_+\), the period is

\[T(E)=2\sqrt{\frac{m}{2}}\int_{x_-}^{x_+}\frac{dx}{\sqrt{E-U(x)}}\]

The factor two covers travel from one turning point to the other and back. This formula assumes one-dimensional bound conservative motion. Near a separatrix or unstable barrier top, the period can grow without bound.

Force and direction at a turning point

At a simple turning point, use

\(F=-dU/dx\)

to decide which velocity branch follows. If force points into the allowed region, motion reverses. At a stationary point where force is also zero, energy alone may describe an unstable equilibrium or an asymptotic approach; a generic sign flip is not justified.

An event-driven trajectory should locate the turning time within a step, set speed to zero within tolerance, choose the branch from local dynamics, and advance any remaining time.

Energy as an oracle for a force integrator

A conventional force-based integrator retains direction naturally. The energy solver can independently check it:

times = np.linspace(0, 3, 101)
positions = np.sin(2 * times)
velocities = 2 * np.cos(2 * times)
energies = 0.5 * velocities**2 + 2 * positions**2
assert np.allclose(energies, 2.0)
print("Maximum energy residual (J):", np.max(abs(energies - 2.0)))

Also compare observed extrema with energy-predicted turning points. Refine the time step and event solve. Agreement in energy alone does not prove correct phase or direction.

Energy projection is a different method

One may rescale velocity after a numerical step so kinetic plus potential equals the target total. This enforces one scalar constraint but changes the computed state and can hide force or integration errors. In multiple dimensions, rescaling preserves current direction even when the correct error has a directional component.

If energy projection is used, name it as part of the algorithm, measure the applied correction, and test its effect on momentum, phase, reversibility, and other invariants. It is not evidence that the unprojected method conserves energy.

Multiple dimensions need more information

In several dimensions,

\[|\mathbf v|=\sqrt{\frac{2[E-U(\mathbf r)]}{m}}\]

defines a magnitude but leaves infinitely many velocity directions. Angular momentum or other constants can reduce possibilities in symmetric problems. For central motion, radial dynamics can use an effective potential

\[U_{\mathrm{eff}}(r)=U(r)+\frac{L^2}{2mr^2}\]

where \(L\) is conserved. The extra term represents transverse kinetic energy, not a new fundamental interaction.

Non-conservative and time-dependent motion

With external work, friction, drag, or time-dependent potential, mechanical energy is no longer one constant input. Evolve an energy ledger:

\[E(t_2)=E(t_1)+W_{\mathrm{noncons}}+W_{\mathrm{external}}\]

Because drag depends on velocity, energy alone may still be insufficient to advance the state. A position declared forbidden by an outdated energy can become reachable after a motor adds work.

Numerical and physical checks

Energy-based solvers can replace part of a trajectory problem with root finding and quadrature. They are most complete in one-dimensional conservative motion, where their lost direction and timing information can be restored with explicit branch and integral rules.

A worked Python notebook

Use total energy 2 joules in a spring with stiffness 4 newtons per metre and mass 1 kilogram. The question for this calculation is precise: how does positive-branch speed change when we change 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.

The negative velocity branch is equally possible; energy alone does not select direction. 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 -0.8 to 0.8 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 position, expressed in m. The output is positive-branch speed, expressed in m/s. 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.

\[f(q) = \sqrt{4 - 4 q^{2}}\]

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 0.5 m, the expression gives 1.73205 m/s. 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("sqrt(4-4*q**2)", locals={"q": q})
reference_input = 0.5
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted positive-branch speed:", reference_output, "m/s")

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 1.2 m/s. The midpoint, at 0 m, gives 2 m/s. The upper endpoint gives 1.2 m/s. Read these together with the assumptions: they describe different choices of 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([-0.8, 0.0, 0.8])
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} m/s")
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 position; the vertical axis shows positive-branch speed. 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(-0.8, 0.8, 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("Position (m)")
ax.set_ylabel("Positive-branch speed (m/s)")
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 positive-branch speed change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.5 m is -1.1547 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.

\[f'(q) = - \frac{4 q}{\sqrt{4 - 4 q^{2}}}\]

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 = 0.508
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")
print("Full expression:", full_prediction, "m/s")
print("Approximation difference:", full_prediction - linear_prediction, "m/s")

Here the changed input is 0.508 m. The tangent estimate is 1.72281 m/s, while the full expression gives 1.72271 m/s. 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 = 1.6
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 position inside the stated interval, predict whether positive-branch speed 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 negative velocity branch is equally possible; energy alone does not select direction. 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: 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(0.8) - evaluate(-0.8))
for points in [101, 201, 401]:
    grid = np.linspace(-0.8, 0.8, 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: