The adiabatic invariant

Slowly tighten the spring under a swinging mass and both its energy and its frequency climb. Is ANYTHING conserved under a slow change of the potential — and is it the same quantity Bohr and Sommerfeld quantized?

Units: dimensionless (R = J_final/J_initial, the ratio of the action J=E/ω before and after a slow change of the potential; the adiabatic theorem predicts R→1. Independent of mass, amplitude, and starting phase)

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The finding

The adiabatic invariant: slowly tightening an oscillator's well raises both its energy and its frequency, yet the action J = E/ω = (1/2π)∮p dq is CONSERVED — recovered from a from-scratch symplectic trajectory as R = J_final/J_initial = 1.000001 over a slow linear ramp k:1→4 (|Δ|=1.4e-6) while the energy RISES by the full frequency ratio √(k₁/k₀)=2.0 (energy is NOT the invariant; the phase-space AREA is), with R=1 NEVER plugged in. |R−1| falls monotonically with ramp duration as a clean power law ~τ⁻² (slowest ramp >1000× more invariant than fastest); a SUDDEN step of the same endpoints breaks it (phase-averaged R=1.25, a deviation 10⁵× the slow ramp's), so slowness is essential; and the action read geometrically as the shoelace phase-space area ∮p dx/2π equals E/ω to 2e-8 — the very ∮p dq Bohr and Sommerfeld quantized (Ehrenfest 1916)

Method

Integrate a from-scratch 1-D oscillator H = ½p² + ½k(t)x² (unit mass, so acceleration a=−k·x and ω=√k) with velocity Verlet (symplectic, so at fixed k the energy stays bounded — the property that makes the measured energy CHANGES under ramping physical rather than numerical). Launch at amplitude 1 with energy E₀=0.5 and SLOWLY ramp the stiffness k from k₀=1 (ω₀=1) to k₁=4 (ω₁=2). Read the action J = E/ω — equivalently the enclosed phase-space area (1/2π)∮p dq — straight off the trajectory's own E(t), ω(t), and track the invariance ratio R = J_final/J_initial, phase-averaged over 24 initial phases. Five probes: (A) a slow linear ramp (τ=400); (B) the energy ratio over that ramp; (C) an adiabaticity scan over τ∈{25…800}; (D) an instantaneous SUDDEN step of the same endpoints; (E) a shoelace measurement of the phase-space loop area held at k₁; plus a smooth-ramp shape-independence check and a slow up-then-down round trip. The value R=1 is loaded from the reference ONLY to score; the recovery sees only the integrated E(t) and ω(t).

The law it recovers

adiabatic invariance of the action, J = (1/2π)∮p dq (Ehrenfest 1916); for the oscillator J = E/ω, and a slow change of k conserves J to all orders in the slowness ε=(1/ω)(dω/dt), even as E and ω individually scale (E_final/E_initial = ω_final/ω_initial = √(k₁/k₀))

Measurements, controls & cross-checks

Slow ramp R

1.000001

Slow ramp abs error

1.4000e-6

Energy ratio

2

Omega ratio

2

Energy not conserved

E rises by √(k₁/k₀)=2.0 while J=E/ω holds — the invariant is the phase-space area, not the energy

Adiabaticity scan

Sudden control

R sudden

1.25

Analytic

(k₀+k₁)/(2k₀)/√(k₁/k₀) = 2.5/2 = 1.25

Deviation vs slow

1.8e5× the slow-ramp deviation

Note

an instantaneous step of the SAME endpoints leaves x,p continuous, so E jumps by ½(k₁−k₀)x² and the action changes — only a SLOW change is adiabatic

Geometric area

Shoelace area over 2pi

0.500049

E over omega

0.500049

Rel diff

2.4000e-8

Note

the conserved action IS the enclosed phase-space area ∮p dq — the Bohr–Sommerfeld quantized quantity (∮p dq = nh)

Shape independence

Smooth ramp R minus 1

1.0800e-10

Linear ramp R minus 1

1.3800e-6

Note

a smooth (C¹) ramp of the same slow τ is even more invariant — J depends only on the endpoints (k₀,k₁), not on the ramp profile, in the adiabatic limit

Round trip

E back

0.9989

J back

0.9989

Note

slow up 1→4 then down 4→1 returns E and J to their starting values — the slow process is reversible

Energy drift fixed k

6.1700e-7

Module honesty

What the screen shows

AdiabaticModule cycles two fully deterministic 1000-frame phases (24 velocity-Verlet substeps of dt=2π/2000 per frame, launch x=1, v=0, J₀=0.5). SLOW: k staircase-ramps 1→4 over 850 frames (updated once per FRAME), then holds — the HUD's R locks at 0.9981. SUDDEN: k jumps at frame 380 — the HUD's R jumps to 1.7951 while quoting the phase-averaged ⟨R⟩ = 1.25.

Bit exact pins

the derisk transcribes _startPhase/_kFor/fixedUpdate verbatim and pins 7 floats strict-=== (R/E/J at slow end 0.9980961675062209/0.9980961675062209/0.49904808375311044, R at hold start 0.9980923869610766, R/E/J at sudden end 1.7951064822854639/1.7951064822854639/0.8975532411427319) plus 3 HUD strings verbatim (slow@999, sudden@999, slow@498). Live headless-browser read-back: 6/6 captures of #phs-adia matched the transcription's checkpoint set byte-identically (slow@588, slow@768, slow@999, sudden@378 ×2, sudden@999).

Disclosure slow

the on-screen slow-ramp R−1 = −1.90e-3 is ~1400× the oracle's τ=400 phase-averaged 1.4e-6 — NOT a contradiction but a different regime: single phase, staircase ramp, τ_mod = 850·24·dt = 64.09. Decomposition (gated): a substep-granularity-k witness moves R−1 to −1.05e-3 and agrees with the oracle's continuous-ramp generator at the same τ and phase (−0.98e-3) to 7.1e-5 ⇒ the frame staircase contributes ≈ −0.86e-3 and the single-phase first-order O(ε) oscillatory term ≈ −0.98e-3; phase-averaging over 24 phases cancels the oscillatory term to +7.06e-5, which lands within 5% of the oracle scan's fitted τ^−2.12 law extrapolated from τ=400 to τ_mod (6.70e-5) — the screen and the oracle measure the same physics at different τ and averaging.

Disclosure sudden

the displayed sudden R = 1.7951 is the SINGLE-trajectory value at the step phase (t = 4.56 periods ⇒ φ = 0.56·2π, R(φ) = (1+3cos²φ)/2 = 1.796726), not the quoted phase-averaged ⟨R⟩ = 1.25 (the HUD labels the 1.25 as phase-averaged, and the 24-phase mean of the step map equals that formula to 1e-15). The remaining −1.62e-3 gap to R(φ) is the module's stale-acceleration transition artifact — the Verlet substep spanning the step averages the old k's acceleration: Δv = ½(k₁−k₀)x·dt, ΔR = (vΔv+½Δv²)/(ω₁J₀) = −1.60e-3 — predicted and gated to |Δ| = 1.7e-5 ≤ 1e-4 (the Verlet phase-error scale).

Hold segment

after the ramp ends (frame 851+) the displayed R is frozen: |R(end)−R(hold start)| = 3.78e-6 ≤ 2e-5 (4× the fixed-k Verlet energy wobble (ω₁dt)²/8) — the 'locks at 1' reading is a real steady value, not an average over drift.

Reference integrity

known_value === 1 exactly (the adiabatic theorem's exact limit value) is itself a gate, so a tampered reference fails even though the recovery never reads it; tamper self-test: known_value 1.0→1.07 fails gates A+I with the recovered 1.000001 unchanged (exit 1); pin tamper (last digit of RslowEnd) fails gate J (exit 1).

Module untouched

true

Note

recovered R = J_final/J_initial = E/ω ratio from the symplectic trajectory; the target R=1 is loaded from the reference only to score. All 19 derisk gates pass (12 physics + 7 module-honesty: reference integrity, bit-exact transcription pins, HUD strings verbatim with live headless read-back, slow-phase staircase/phase-averaging decomposition, sudden-phase single-trajectory reconciliation with the stale-acceleration artifact predicted analytically, hold-segment freeze).

What it reduces to

The adiabatic invariance of the action, J = (1/2π)∮p dq, first proved for the general slowly-perturbed periodic system by Ehrenfest (1916) building on Boltzmann and Clausius's mechanical heat theorem; for the harmonic oscillator J = E/ω. This world VALIDATES, not derives: it assumes only Newtonian dynamics (a=−k(t)x integrated symplectically) and recovers the invariance ratio R=J_final/J_initial=1 as a pure number, with R=1 never entering the recovery — only the trajectory's E(t),ω(t) are read. It is non-circular on several fronts — the recovery sees only E and ω; the invariance holds while the energy manifestly does NOT (E rises by the full frequency ratio √(k₁/k₀)=2, isolating the phase-space AREA as the conserved quantity, not the naive energy); the adiabaticity scan gives the falsifiable perturbation prediction that slower ⇒ more invariant with a clean ~τ⁻² law; the SUDDEN-step control shows a fast change of the same endpoints breaks the invariant (R=1.25), proving slowness is essential; and an independent shoelace measurement confirms the conserved number IS the enclosed phase-space area to 2e-8. It is DISTINCT from the virial theorem (?world=virial, a time-average of a single fixed orbit) and from resonance/coupled oscillators (?world=resonance, ?world=coupled, fixed-parameter response): here the parameter itself is slowly driven and the conserved object is the action of the slowly-deforming orbit. Its deepest significance is the bridge to old quantum theory: this ∮p dq is exactly the quantity Bohr and Sommerfeld quantized (∮p dq = nh), so its classical adiabatic invariance is why a quantum system's quantum number is preserved under a slow perturbation — the classical seed of the quantum adiabatic theorem (Born–Fock 1928) and of Berry's phase. It does not treat non-integrable/chaotic adiabatic invariants (where invariance can fail at separatrix crossings), multiple slowly-varying parameters, or the quantum problem itself; it establishes the cleanest classical case, the oscillator whose action is E/ω. The lab's first adiabatic-invariant world.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

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

Oracle

scripts/oracles/adiabatic.reference.json

Sources

P. Ehrenfest, 'Adiabatische Invarianten und Quantentheorie', Ann. Phys. 356, 327 (1916). L. D. Landau & E. M. Lifshitz, 'Mechanics' §49 (the adiabatic invariant J=(1/2π)∮p dq; I=E/ω for the oscillator). A. Sommerfeld, 'Atombau und Spektrallinien' (∮p dq = nh). M. Born & V. Fock, Z. Phys. 51, 165 (1928) — the quantum adiabatic theorem. Reference value: R = J_final/J_initial = 1 (adiabatic limit); R_sudden = (k₀+k₁)/(2k₀)/√(k₁/k₀) = 1.25 for k₀=1,k₁=4.