Wavelace

Lagrange Points

Bodies · mechanical systems

restricted
m=1-μ/100
m=μ/100
x=1-μ/100-cbrt(μ/300)
x=1-μ/100+cbrt(μ/300)
x=-1-5μ/1200
x=0.5-μ/100  y=sqrt(3)/2
x=0.5-μ/100  y=-sqrt(3)/2

Open in the app The dials and keys named below are the app's.

The setup

Two heavy bodies stand one unit apart, with masses 0.99 and 0.01 at the opening value μ = 1, the lighter on the right. The frame turns with them, once every 2π ≈ 6.28 s at Speed 1, so they stand still on screen. Five massless particles sit at the five places where a particle turning with the pair feels no net force. L1 is between the two bodies, L2 beyond the small one and L3 on the far side of the heavy one. L4 and L5 sit at the corners of the two equilateral triangles on the pair, L4 ahead of the small body and L5 behind it.

Three let go, two hold

L1, L2 and L3 sit on saddles of the turning frame's potential, so any offset grows. They are placed by a formula good to first order in the mass ratio, which leaves them 0.009, 0.006 and 0.0002 off, and those offsets are what grow. L1 is 0.5 off its place at 1.8 s, L2 at 15 s, L3 at 31 s. L4 and L5 sit on hilltops, yet they hold: the Coriolis term turns a particle sliding off into a small loop around its point. In two minutes they stray less than 0.007.

The threshold

The triangle points hold only while the small body is light enough. The mass ratio must be under about 0.0385, or 3.85 on the slider. At 3.8 the two stay within 0.027 of their places for two minutes, and at 3.9 they drift 0.19 in that time. At 5 they are gone after 30 s, at 10 after 18 s. Jupiter is a thousandth of the Sun, well inside the limit, and thousands of Trojan asteroids share its orbit at its L4 and L5.

History

Leonhard Euler found the three points on the line in 1767, and Joseph-Louis Lagrange the two triangle points in 1772. Edward Routh gave the mass limit for their stability in 1875.

Try

  • Drag μ to 5: L4 and L5 hold for a while and then wander off after about 30 s. At 10 they go after 18 s.
  • Drag μ to 0.1: the small body is a thousandth of the pair, L1 and L2 still go within 4 s, and L3 lasts 74 s.
  • Set Speed to 4: two minutes play in 30 s, so L3 lets go in under 8 s of your time.
  • Top in the deck shows the five places flat, the pair on a diagonal and the two triangles on it.

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Bodies · mechanical systems