The virial theorem reads a force law's exponent off a single orbit

Kepler's third law reads a period off an orbit. Can a SINGLE bound orbit read off the EXPONENT of the force law that made it — the −1 of gravity, the +1 of a linear pull, the +2 of a spring — from nothing but its time-averaged kinetic and potential energies?

Units: dimensionless (n = homogeneity/scaling degree of U(r) ∝ rⁿ: gravity/Kepler −1, linear +1, harmonic +2; 2⟨T⟩/⟨U⟩ is a ratio of energies, independent of mass, coupling, and orbit size)

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The finding

The virial theorem reads a force law's exponent off a single orbit: for U ∝ rⁿ the pure number 2⟨T⟩/⟨U⟩ equals n — recovered from a from-scratch symplectic orbit as −1.00000 for gravity (Kepler, 2⟨T⟩=−⟨U⟩), +0.9999 for a linear confining potential, +2.0000 for a harmonic spring (a slope-1 line through n_true vs n_recovered), with n NEVER plugged in. The INSTANTANEOUS ratio ⟨2T/U⟩ MISSES n (−0.82 not −1, 3.80 not 2) — only the ratio of separately time-AVERAGED energies is the invariant; the recovered n is eccentricity-INDEPENDENT (−1 across e=0…0.85), and a mixed non-power-law potential gives a non-integer, orbit-dependent ratio, so the clean integers require a pure power law (Clausius 1870)

Method

Integrate a from-scratch central-force orbit around a fixed mass with velocity Verlet (symplectic, so total energy stays bounded over hundreds of orbits — the property that makes time-averages trustworthy). Launch from perihelion (position (r_min,0), velocity (0,v_peri), so the virial G=m·r·v starts at 0) and time-average the kinetic energy ⟨T⟩=⟨½v²⟩ and potential energy ⟨U⟩ over 300 orbits. The virial theorem predicts that for a homogeneous potential U ∝ rⁿ the pure number 2⟨T⟩/⟨U⟩ equals the exponent n. Measure it for three pure power laws — gravity U=−μ/r (n=−1, Kepler), linear U=k·r (n=+1), harmonic U=½k·r² (n=+2) — and fit a line through (n_true, n_recovered). Four controls: (A) the Kepler orbit-average identity 2⟨T⟩+⟨U⟩=0; (B) the INSTANTANEOUS pointwise ratio ⟨2T/U⟩, which is NOT the virial combination; (C) an eccentricity scan e=0…0.85; (D) a MIXED non-power-law potential U=−A/r+½Br². The integers n_true are loaded from the reference ONLY to score; the recovery sees only the trajectory's T(t) and U(t).

The law it recovers

virial theorem 2⟨T⟩ = ⟨r·∇U⟩ = n⟨U⟩ for U∝rⁿ (Euler homogeneity); for gravity n=−1 ⇒ 2⟨T⟩=−⟨U⟩, and the orbit-averaged energies are ⟨T⟩=−E, ⟨U⟩=2E over a Kepler period

Measurements, controls & cross-checks

Recovered gravity n

-1

Recovered gravity abs error

2.6000e-6

Recovered linear n

0.99989

Recovered harmonic n

2

Family slope

1

Kepler identity

Two T plus U

-2.6000e-6

Note

2⟨T⟩+⟨U⟩ ≈ 0 to machine level — ⟨T⟩=0.5000=|E|, ⟨U⟩=−1.0000=2E for the a=1,e=0.6 Kepler orbit

Family

Instantaneous control

Gravity instant mean

-0.82

Gravity true

-1

Miss

1.8

Harmonic instant mean

3.8

Harmonic true

2

Note

the mean of the pointwise ratio 2T(t)/U(t) MISSES the exponent (−0.82 vs −1, 3.80 vs 2) because T and U each swing by large factors around the orbit; only 2⟨T⟩/⟨U⟩, the ratio of the SEPARATELY time-averaged energies, is the virial invariant

Eccentricity independence

Mixed control

Orbit 1

-1.338

Orbit 2

-1.552

Spread

0.214

Note

a non-power-law potential U=−A/r+½Br² gives a non-integer 2⟨T⟩/⟨U⟩ that CHANGES from orbit to orbit — so the clean integers −1/+1/+2 are a genuine signature of a pure power law, not an artifact of the averaging

Energy drift max

1.1000e-6

Note

recovered n = 2⟨T⟩/⟨U⟩ from the symplectic orbit; the true homogeneity degree is loaded from the reference only to score. All 16 derisk gates pass.

What it reduces to

The virial theorem of Clausius (1870): for a bound orbit the virial G = Σpᵢ·rᵢ is bounded, so ⟨dG/dt⟩→0, and since dG/dt = 2T − r·∇U this gives 2⟨T⟩ = ⟨r·∇U⟩; for a homogeneous potential U∝rⁿ, Euler's theorem makes r·∇U = n·U, so 2⟨T⟩ = n⟨U⟩ and n = 2⟨T⟩/⟨U⟩. This world VALIDATES, not derives: it assumes only Newtonian dynamics (F=−∇U integrated symplectically) and recovers the potential's scaling exponent as a pure number from two time-averaged energies, with n never entering the recovery. It is non-circular on several fronts — the recovery reads only T(t),U(t); it succeeds for three independent power laws with a slope-1 fit (the perturbation prediction that the exponent tracks the force law); it is eccentricity-independent (a force-law property, not an orbit-shape coincidence); the instantaneous-ratio control shows the naive pointwise average FAILS, isolating the virial combination as the actual invariant; and the mixed-potential control shows the clean integers demand a pure power law. It is DISTINCT from ?world=kepler (which recovers the third-law scaling T²∝a³ between DIFFERENT orbits): the virial theorem is a statement about the time-average of a SINGLE orbit, and its gravitational case 2⟨T⟩=−⟨U⟩ is the foundation of the virial mass estimates that first revealed dark matter (Zwicky 1933, the Coma cluster) — a galaxy cluster's total mass is read from the time-averaged kinetic and potential energies of its members exactly as this orbit's exponent is. It does not treat self-gravitating N-body relaxation, non-closed relativistic orbits, or quantum expectation-value virials; it establishes the classical virial invariant on the cleanest possible system. The lab's first virial-theorem world.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- virial (scripts/virial-derisk.mjs)

Oracle

scripts/oracles/virial.reference.json

Sources

R. Clausius, 'Über einen auf die Wärme anwendbaren mechanischen Satz', Ann. Phys. 217, 124 (1870) — the virial theorem. H. Goldstein, C. Poole, J. Safko, 'Classical Mechanics' 3rd ed. §3.4. L. D. Landau & E. M. Lifshitz, 'Mechanics' §15 (2⟨T⟩=−⟨U⟩ for the Kepler problem). F. Zwicky, Helv. Phys. Acta 6, 110 (1933) — the virial mass of the Coma cluster. Reference exponents: n_gravity=−1, n_linear=+1, n_harmonic=+2.