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Many-body systems

A many-body system contains interacting parts whose individual motions may be complicated even when the system obeys simple conservation laws. Planets, gas molecules, charged particles, grains, and colliding carts all fit this pattern at different scales. The central modeling decisions are what counts as a body, which interactions are retained, and where the system boundary is drawn.

Describing the state

For \(N\) point particles with fixed masses, a classical state at one instant contains positions and momenta:

\[\{\mathbf r_i,\mathbf p_i\}_{i=1}^{N}\]

Masses and force-law parameters also belong to the model data. Identity matters when bodies have different properties or when histories must be tracked. Derived values such as total momentum and center of mass should be calculated from the authoritative state, not updated independently where they can drift out of agreement.

import numpy as np

masses = np.array([1.0, 2.0, 1.0])
positions = np.array([[0.0, 0.0], [1.2, 0.0], [0.0, 1.2]])
momenta = np.zeros_like(positions)
print("Number of bodies:", len(masses))

For extended rigid bodies, orientation and angular momentum are also needed. Deformable bodies need still more internal coordinates or a field description. Calling every object a point particle deliberately discards size, rotation, collision shape, and internal energy.

Equations of motion

For fixed-mass particles in an inertial frame,

\[\frac{d\mathbf r_i}{dt}=\frac{\mathbf p_i}{m_i},\qquad \frac{d\mathbf p_i}{dt}=\mathbf F_i\]

The total force can be separated into external and internal parts:

\[\mathbf F_i=\mathbf F_i^{\mathrm{ext}}+\sum_{j\ne i}\mathbf F_{ij}\]

Here \(\mathbf F_{ij}\) means the force on body \(i\) due to body \(j\). The index convention must be stated; reversing it silently reverses action and reaction.

For pair interactions satisfying

\[\mathbf F_{ij}=-\mathbf F_{ji}\]

the internal forces cancel when summed. This produces the system momentum balance

\[\frac{d\mathbf P}{dt}=\sum_i\mathbf F_i^{\mathrm{ext}},\qquad \mathbf P=\sum_i\mathbf p_i\]

Equal-and-opposite pair forces are sufficient for this classical result. Angular momentum conservation additionally requires the internal forces to create zero net internal torque, as central pair forces do.

Calculating every pair once

For symmetric pair interactions, visit each unordered pair once. Calculate both force contributions from the same relative position and add them to separate accumulators. This avoids double counting and prevents one side from being evaluated with a different rounded distance.

def pair_forces(positions, stiffness=2.0, rest_length=1.0):
    forces = np.zeros_like(positions)
    for i in range(len(positions) - 1):
        for j in range(i + 1, len(positions)):
            separation = positions[j] - positions[i]
            distance = np.linalg.norm(separation)
            if distance == 0:
                raise ValueError("Overlapping nodes have no spring direction.")
            force = stiffness * (distance - rest_length) * separation / distance
            forces[i] += force
            forces[j] -= force
    return forces


print("Forces (N):", pair_forces(positions))

The number of unordered pairs is

\[N_{\mathrm{pairs}}=\frac{N(N-1)}{2}\]

so a direct all-pairs calculation grows quadratically. Neighbor lists can accelerate short-range interactions. Tree, multipole, or mesh methods can approximate long-range forces. Each approximation introduces parameters and errors that require convergence checks.

System-level reductions

Total mass, center of mass, momentum, and kinetic energy are

\[M=\sum_i m_i,\quad \mathbf R_{\mathrm{cm}}=\frac{1}{M}\sum_i m_i\mathbf r_i,\quad \mathbf P=\sum_i\mathbf p_i,\quad K=\sum_i\frac{p_i^2}{2m_i}\]

If a conservative pair potential \(U_{ij}\) is used, total potential energy counts each unordered pair once:

\[U=\sum_{i<j}U_{ij}+\sum_iU_i^{\mathrm{ext}}\]

Summing over both \((i,j)\) and \((j,i)\) without a factor of one-half would double the pair energy. Force and energy implementations should be cross-checked by verifying that force equals the negative gradient of the same potential.

Center-of-mass and relative motion

The center of mass obeys

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

for a fixed-mass system. Internal interactions can rearrange bodies, heat the system, or produce collisions without changing the motion of the center of mass. This provides a simple check on otherwise complex motion.

For two bodies, center-of-mass and relative coordinates separate the translational and internal parts:

\[\mathbf R=\frac{m_1\mathbf r_1+m_2\mathbf r_2}{m_1+m_2},\qquad \mathbf r=\mathbf r_1-\mathbf r_2\]

The kinetic energy becomes

\[K=\frac12M V_{\mathrm{cm}}^2+\frac12\mu v_{\mathrm{rel}}^2\]

with reduced mass \(\mu=m_1m_2/(m_1+m_2)\). This explains why collision and orbital calculations often become simpler in the center-of-momentum frame.

Boundaries and external interactions

Whether a force is internal depends on the selected system. Gravity from Earth is external for a system containing only a thrown ball, but internal for a system containing ball and Earth. A wall impulse changes the momentum of a particle-only system while the larger wall-Earth system receives the opposite momentum.

Mass flow also crosses boundaries. A rocket that excludes its exhaust is a changing-mass open system, so summing the rocket's momentum alone does not produce a constant. Record which bodies enter or leave and the momentum they carry.

Periodic boundaries need similar care. A particle crossing one edge may reappear at the opposite edge without a physical jump. Wrapped coordinates are suitable for neighbor searches, while unwrapped coordinates may be needed for displacement and center-of-mass calculations.

Singular and short-range interactions

Newtonian point gravity and Coulomb forces diverge as separation approaches zero. That mathematical singularity can make a point-particle model and a finite-step numerical method fail during close encounters. Possible treatments include resolving the encounter with adaptive steps, modeling finite radii and collisions, or introducing a documented softening length.

Softening changes the physical force law, especially at short range. It is not merely a display choice. Results must be tested as the softening length changes and interpreted only at scales where the modified model is acceptable.

Cutting off a pair force abruptly can make potential energy discontinuous and can inject artifacts when pairs cross the cutoff. Shifted or smoothly tapered potentials can reduce the discontinuity, but they too define a modified model.

Advancing a shared snapshot

All forces for one step should be calculated from a consistent state. Updating body 1's position before computing the force involving body 2 mixes times and makes the result depend on array order. Use force arrays or a suitable staged integrator.

dt = 0.01
forces = pair_forces(positions.copy())
next_momenta = momenta + dt * forces
next_positions = positions + dt * next_momenta / masses[:, None]
assert np.allclose(next_momenta.sum(axis=0), momenta.sum(axis=0))
print("Next positions (m):", next_positions)

For conservative long-time motion, method choice matters. A symplectic integrator often preserves geometric structure better than a generic explicit method, but no method removes the need to reduce the time step and compare at the same physical time.

Determinism, ordering, and floating point

Mathematical addition is associative; floating-point addition is not exactly so. Reordering bodies or parallel reductions can change final low bits and, in chaotic systems, those small differences can grow into different detailed trajectories. This does not automatically indicate different statistical or conserved behavior.

Use stable identifiers, deterministic pair ordering when exact reproducibility matters, and pairwise or compensated summation for large totals. Assess invariants and statistical quantities rather than demanding bitwise agreement from chaotic long-time states.

Verification strategy

Start with one body under a known external force, then two isolated bodies with a central interaction. Confirm equal-and-opposite pair forces, constant total momentum, uniform center-of-mass motion, and expected energy behavior. Translate and rotate the entire initial state and verify that the result transforms correspondingly when the force law has those symmetries.

Then test three or more bodies, permute storage order, and compare reductions. Refine time step, force approximation, cutoff, and spatial resolution separately. A flat total momentum trace does not prove positions are correct, so compare analytic special cases and independently calculated accelerations too.

Reliability checks

Many-body reasoning becomes manageable when local interactions and global balances remain visible at the same time. Individual trajectories may be complicated, but the center of mass, momentum, energy ledger, and symmetry tests provide strong anchors for checking the model.

A worked Python notebook

Inspect one pair contribution for masses 1 and 2 in units where G equals 1. The question for this calculation is precise: how does pair gravitational potential change when we change pair separation, 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 complete many-body potential sums each unordered pair exactly once. 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 0.5 to 4 scaled length units. 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 pair separation, expressed in scaled length units. The output is pair gravitational potential, expressed in scaled energy units. 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.

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

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 2 scaled length units, the expression gives -1 scaled energy units. 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", locals={"q": q})
reference_input = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "scaled length units")
print(
    "Predicted pair gravitational potential:", reference_output, "scaled energy units"
)

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 scaled energy units. The midpoint, at 2.25 scaled length units, gives -0.888889 scaled energy units. The upper endpoint gives -0.5 scaled energy units. Read these together with the assumptions: they describe different choices of pair separation 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([0.5, 2.25, 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} scaled length units -> "
        f"{output_value:.6g} scaled energy units"
    )
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 pair separation; the vertical axis shows pair gravitational potential. 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(0.5, 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("Pair separation (scaled length units)")
ax.set_ylabel("Pair gravitational potential (scaled energy units)")
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 pair gravitational potential change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 scaled length units is 0.5 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.

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

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 = 2.0175
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, "scaled energy units")
print("Full expression:", full_prediction, "scaled energy units")
print(
    "Approximation difference:",
    full_prediction - linear_prediction,
    "scaled energy units",
)

Here the changed input is 2.0175 scaled length units. The tangent estimate is -0.99125 scaled energy units, while the full expression gives -0.991326 scaled energy units. 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 = 3.5
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 pair separation inside the stated interval, predict whether pair gravitational potential 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 complete many-body potential sums each unordered pair exactly once. 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 scaled energy units: 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(0.5))
for points in [101, 201, 401]:
    grid = np.linspace(0.5, 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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