Analytical test cases validation - progenax — differentiable initial conditions for N-body simulations

progenax’s analytical IC builders (progenax.analytical) emit exact-solution initial conditions whose defining invariants can be checked directly. Test file: tests/validation/test_analytical_physics.py (see the test dashboard for the live per-suite count); figures: scripts/validate_analytical.py. The orbits are integrated in-script with symplectic velocity-Verlet (the scheme the tests use); the oracle is always the analytic invariant — the energy formula, the figure-eight’s $L=0$ and closure, Kepler’s third law, the harmonic solution — not the integrator.

What is verified¶

Rows map to test_analytical_physics.py; Measured values are regenerated by scripts/validate_analytical.py.

Property	Tolerance (as tested)	Measured	Anchor
Two-body energy $E = -Gm_1m_2/2a$	$< 10^{-12}$ (indep. of $e$ )	exact (-0.0500)	Kepler / vis-viva
Two-body Kepler III period	$< 10^{-10}$ rel	exact	$T = 2\pi\sqrt{a^3/GM}$
Eccentric ellipse closes ( $e=0.5$ )	$< 10^{-4}$ after 1 period	$1.1\times10^{-7}$	bound Kepler orbit
Figure-eight total $L$	$< 10^{-10}$	0 (exact)	Chenciner–Montgomery $L=0$
Figure-eight closes	$< 10^{-6}$ after 1 period	$4.3\times10^{-8}$	choreography periodicity
Figure-eight COM/momentum	0 to 10^-8	$\le10^{-8}$	barycentric, $\sum m\mathbf v=0$
Kepler III across 8 planets	rel $< 10^{-2}$ vs observed $T$	max $6.9\times10^{-3}$	JPL Horizons $a$ + observed $T$
Planet mass ratios (IAU 2009)	rel $< 2\times10^{-3}$	332945 / 1047.4 / 3498	Earth / Jupiter / Saturn
Harmonic $x(t)=A\cos(\omega t+\phi)$	max dev $< 10^{-3}$	$5.7\times10^{-7}$	SHM exact solution
Harmonic energy $E=\tfrac12 m\omega^2A^2$	$\Delta E/E < 10^{-4}$	$5.6\times10^{-7}$	SHM invariant
Adversarial: spun figure-eight fails	$L\neq0$ and no closure	$L=0.30$ , closure 2.1	wrong IC correctly rejected

Figures¶

Generated by scripts/validate_analytical.py (PASS/FAIL per panel; PNG + PDF vector).

The Chenciner–Montgomery figure-eight. Three equal masses chase one another along a
single figure-eight curve (grey); the markers are the canonical collinear start. The
total angular momentum is exactly zero and the orbit closes after one period to
4\times10^{-8} — the two defining properties of the choreography. — Figure 1:**The Chenciner–Montgomery figure-eight.** Three equal masses chase one another along a single figure-eight curve (grey); the markers are the canonical collinear start. The total angular momentum is **exactly zero** and the orbit closes after one period to $4\times10^{-8}$ — the two defining properties of the choreography.

Two-body Kepler. (a) An e=0.5 relative orbit integrated one period returns to its
start (residual 1.1\times10^{-7}), with the focus at the COM. (b) Every invariant is
well inside tolerance: the energy equals -Gm_1m_2/2a to machine precision, and
\Delta E/E and \Delta|L| over the period are far below their thresholds. — Figure 2:**Two-body Kepler.** (a) An $e=0.5$ relative orbit integrated one period returns to its start (residual $1.1\times10^{-7}$ ), with the focus at the COM. (b) Every invariant is well inside tolerance: the energy equals $-Gm_1m_2/2a$ to machine precision, and $\Delta E/E$ and $\Delta|L|$ over the period are far below their thresholds.

Kepler’s third law across the solar system. (a) The eight planets (JPL a, observed
sidereal T) lie on T^2 = (4\pi^2/GM_\odot)\,a^3 over three decades in a^3. (b) The
Kepler-predicted period matches the observed period to \le0.7\% for every planet (the
residual is dominated by orbital perturbations, not the builder). The planet/Sun mass
ratios match the IAU 2009 values to \le10^{-4}. — Figure 3:**Kepler’s third law across the solar system.** (a) The eight planets (JPL $a$ , observed sidereal $T$ ) lie on $T^2 = (4\pi^2/GM_\odot)\,a^3$ over three decades in $a^3$ . (b) The Kepler-predicted period matches the observed period to $\le0.7\%$ for every planet (the residual is dominated by orbital perturbations, not the builder). The planet/Sun mass ratios match the IAU 2009 values to $\le10^{-4}$ .

Simple harmonic oscillator. (a) The integrated trajectory tracks the exact
x(t)=A\cos(\omega t+\phi) to 6\times10^{-7}. (b) The energy stays at
\tfrac12 m\omega^2 A^2 to \Delta E/E \approx 6\times10^{-7} over a period. — Figure 4:**Simple harmonic oscillator.** (a) The integrated trajectory tracks the exact $x(t)=A\cos(\omega t+\phi)$ to $6\times10^{-7}$ . (b) The energy stays at $\tfrac12 m\omega^2 A^2$ to $\Delta E/E \approx 6\times10^{-7}$ over a period.

Why the IC must be exact (adversarial). (a) The canonical figure-eight closes with
L=0. (b) A plausible mistake — adding a spurious solid-body rotation to the velocities
— injects net angular momentum (L=0.30\neq0) and shatters the choreography: the three
bodies drift apart and never close. This is the failure mode the closure + L=0 tests
are designed to catch. — Figure 5:**Why the IC must be exact (adversarial).** (a) The canonical figure-eight closes with $L=0$ . (b) A plausible mistake — adding a spurious solid-body rotation to the velocities — injects net angular momentum ( $L=0.30\neq0$ ) and shatters the choreography: the three bodies drift apart and never close. This is the failure mode the closure + $L=0$ tests are designed to catch.

How to run¶

# exact-solution physics tests (~2 s on CPU)
pytest tests/validation/test_analytical_physics.py -q

# regenerate the five figures with PASS/FAIL tables
python scripts/validate_analytical.py

What this suite does not test¶

Long-term integrator stability — progenax’s analytical builders only provide ICs; long-term N-body integration and its conservation are gravax’s responsibility. (The in-script Verlet here runs for a single period purely to expose the invariants.)
Relativistic corrections — Newtonian gravity only.

References¶

Chenciner & Montgomery (2000), Ann. Math. 152, 881 and Simó (2001) for the figure-eight; JPL Horizons (J2000.0) and the IAU 2009 nominal masses for the solar system; standard celestial-mechanics texts for Kepler and the harmonic oscillator. The builders are in progenax/analytical/.