Two masses orbit their common centre on circles. In a frame rotating with them, a third, tiny body feels gravity from both plus the centrifugal effect, which combine into an effective potential Ω = −(1−μ)/r₁ − μ/r₂ − ½(x² + y²) in units where R = 1, total mass = 1 and ω = 1. The surface you see is Ω (clipped near the masses); the five flat spots are the Lagrange points, where all forces balance.

• L1, L2, L3 lie on the line through the masses and are found by solving ∂Ω/∂x = 0 with Newton's method (exact to machine precision, not the cube-root approximation). They are saddle points — unstable, which is why JWST (Sun–Earth L2) and SOHO (L1) fly station-keeping halo orbits.
• L4, L5 form equilateral triangles with the two masses. They are hilltops of Ω, yet the Coriolis force makes them stable when μ < 0.03852 (Routh's criterion, M₁/M₂ ≳ 25) — home to Jupiter's Trojan asteroids.
• Hill sphere ≈ R ∛(μ/3): roughly where M₂'s gravity dominates for satellites.
• Period: T = 2π √(R³ / (G(M₁+M₂))), G = 6.6743×10⁻¹¹.

Test particles are integrated in the rotating frame (including Coriolis) with RK4. Real Sun–Earth μ ≈ 3×10⁻⁶ makes the Earth's well invisible, so the picture exaggerates μ by default; all numbers always use the true masses. Sun–Earth L1/L2 come out ≈ 1.49–1.50 million km from Earth.