In a two-body system there are five points where a small body can sit still in the rotating frame — gravity and centrifugal force cancel. L1, L2, L3 lie on the line (unstable saddles); L4 and L5 sit 60° ahead and behind, at the tips of equilateral triangles, and are stable if the mass ratio exceeds 24.96. Jupiter’s Trojan asteroids live at its L4/L5; JWST sits at Sun–Earth L2.
THE TECHNIQUE 5 points; L4/L5 at 60°, stable if M₁/M₂ > 24.96
The demo checks the Sun–Earth ratio (333,000) against the stability threshold and the 60° geometry: live demo
HISTORY & CREDIT Euler & Lagrange · 1760s-1772
“All five Lagrange points are stable.” — only L4 and L5 can be stable (and only if M₁/M₂ > 24.96); L1–L3 are saddles needing station-keeping. cited
the cancel · in the rotating frame, gravity + centrifugal = 0 at five spots. the triangles · L4, L5 at 60° — stable above the 24.96 ratio. 1772 · Joseph-Louis Lagrange (L4/L5); Leonhard Euler found the collinear ones first.
Five still points in a spinning frame. physics
RECOMMEND FOR I-13 stability, on the compiler
On i-13, Sun–Earth (ratio 333,000 > 25) has stable L4/L5 at 60°:
$ i13 run gw_lagrange.i13
RUN OK · 22 step(s)
points = 5
l4_angle = 60 l5_angle = 60
mass_ratio = 333000 (> 24.96)
l4_l5_stable = 1 triangular = 1
Recommend as a NULL — a theorem (B39). The five equilibria and the 24.96 stability bound are exact results of the restricted three-body potential. A fact, not a same-function difference. NULL — five places to park.