The effective-potential generator
The combined gravitational and centrifugal potential along the line joining two bodies in the frame that rotates with them. Its three stationary points are the collinear Lagrange points, and all three are maxima along this line.
3 essays call
effective-potential. The drawing above is what it returns with no arguments at all; every
call below passes it something, because a placement that passes nothing draws whichever member
of the family the generator happens to default to rather than the one its essay argues about.
Where it is called
Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.
Gravitation
Five places that keep station, in a problem with no solution
Three bodies under gravity cannot be solved. Restrict the problem slightly and five exact answers fall out anyway — three of them roots of a quintic, two of them perfect equilateral triangles.
Gravitation
Three bodies, and what "no solution" actually means
The three-body problem is routinely called unsolvable. Trajectories are computed for it every day, exact periodic solutions are known, and both statements are true — the word is doing more work than it looks.
Gravitation
The curve that says where a body cannot go
The restricted three-body problem has no solution and one conserved quantity. That quantity is enough to draw a boundary the body can never cross — without integrating anything, without knowing where it started, and for all time.