Where are the L1, L2, and L3 points located?

[Stargate38]

I'm trying to figure out where the L1, L2, and L3 points are for a given f(r) and masses M and m. So far I've found this, which shows how L1 is calculated, even though it's only for 1/r^2. At the bottom, it says how to calculate L2 with a slight change in equation:

http://www.phy6.org/stargaze/Slagrang.htm

Can someone please tell me how to convert these equations to a different gravity law, such as r^n (n<1) or 10^(3-3*r)+a/r^n? Are they based on v(r)=sqrt(G*M*f(r)*r) and p(r)=2*pi*r/v(r)=2*pi*sqrt(r/(G*M*f(r)))? I've been trying over and over to find the L2 point for 1/r, but my sims have been falling apart (objects either collide or fly apart). I already know where L4 and L5 are (they form equilateral triangles with the primary and secondary), but I want to be able to create good LPoint sims for all the points in any given law, using a generalized equation based on f(r) and the 2 masses M and m.

P.S.: I know that for r^0, the Lagrange points are easy to calculate, based on the values of M and m, and that no matter what the law is, the L1 point IS the barycenter if M=m, regardless of their orbital velocity. Also, for r^n with n>1, L1-L3 disappear completely and L4, L5 become unstable.

[testtubegames]

I had to go back in my notes to check and see how I calculated this the last time it came up. Conceptually, it isn't too hard, though the math gets pretty tricky to solve. (That's what wolfram alpha is for, right?)

The link you posted does a pretty good job of explaining what's going on. Basically, you have two equations: one tells you the speed you need to go to move in a circle around a given force... and the second tells you the force the object feels:

1. a = v^2/r
2a. F/m = GM1/(r1)^2 +- GM2/(r2)^2

And for our purposes, that second one can be extended to:

2b. F/m = GM1*f(r1) +- GM2*f(r2)

The plus/minus just depends on whether you're look for a lagrange point between the objects, or not between the objects. M1 is the star's mass, M2 is the planets mass, m is the lp objects mass (which doesn't matter of course). r1 is the distance from the star to lp, and r2 is the distance from the planet to the lp.

So, for f(r) = 1/r, we'd have:

2c. F/m = GM1/r1 +- GM2/r2

Suppose we're looking for L2, as you say. Then we use the plus sign (since both the planet and star are pulling the same way). One extra equation that we need is for the angular speed of an object:

3. w = v/r

The angular speed is the linear speed divided by the distance. Since the lp and the planet need to be going around the star at the same angular speed, we can figure out the acceleration the lp needs to have:

acceleration of planet = GM1*f(r_planet_to_star)

which according to equation (1), means the velocity of the planet is:

v_planet = sqrt(r_planet_to_star*GM1*f(r_planet_to_star))

and since we're matching angular velocities, we know that:

v_lp = v_planet*r1/r_planet_to_star

And from that (with equation 1) we can find the acceleration, which we need to match to 2c. And to avoid making this already-long post even longer, I'll leave it there. But hopefully that gives you the right idea for how to approach the problem.