The more complex material which now appears in the section Framed below was originally developed for another
means of delivery.
It can better be read directly, in The Lagrange Points Reconsidered.
This page presents the core of that material.
For three gravitating bodies in constant-pattern motion, the Lagrange Points are where the negligible-mass body can be.
For three bodies affected only by mutual gravity and in constant-pattern motion, the Lagrange Points are positions of the negligible-mass body.
The Lagrange Points are the possible positions of a light particle orbiting in a constant pattern with two heavy bodies.
At a Lagrange Point, the gravitational fields from two free massive bodies give the inwards field needed for a third light body to maintain a constant relative position.
The Lagrange Points are the positions of the lightest body in constant-pattern solutions of the three-body problem, when its mass is negligible.
The Lagrange Points are places where the gravity of two bodies can maintain another body in a constant relative position.
History is now in a separate page : The History of the Lagrange Points. The history is not as has commonly been believed.
The methods in this page and in the framed page follow Lagrange in using the distances between the bodies, but are otherwise independent.
To Be Emptied
To Be Merged
Consider an isolated system of bodies and particles as in Lagrange's Essai, but using little or no algebra. The only interactions are their mutual gravitational attractions. The gravitational field of a body is proportional to its mass, diminishes with distance, and imparts a corresponding component of acceleration to another body along the line joining them.
Let there be two separate massive point bodies "M" and "m". Their barycentre "B" will always be on the line between M and m, and the ratio of its distances from M and m will be fixed (in inverse proportion to their masses). There are no external forces, so we may treat B as stationary. The bodies will form, with B, a constant pattern, which must change in size and/or orientation. As there are no transverse forces, the bodies will move in a plane or a line.
Let there also be independent particles (point bodies of infinitesimal mass) moving nearby. The bodies affect the particles, but not vice versa. Consider whether and how the particles, given suitable relative positions and velocities, can continue to be part of the constant pattern.
If M and m have angular momentum about B, a plane is defined. If a particle is not in that plane, there will be a net component of gravitational field towards that plane, and the pattern cannot then be constant. So it is only necessary to consider possible motions of particles within that plane. If there is no angular momentum, all planes containing the line are equivalent.
For the pattern to be maintained, the acceleration of, and therefore the net field at, each mass and particle must be radial and proportional to its distance from B. The bodies comply, since the field at each is due to the mass of the other, and theit position and motion implies the constant of proportionality.>
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Firstly, consider the three segments of the infinite straight line separated by M and m.
For the outer two segments, the net field is everywhere towards B. It is zero at infinity, and is infinite at the terminating body. It must, therefore, have the right value somewhere in between. Therefore L2 and L3 exist on the line, outside the bodies.
In the shorter part BM of the central segment, the net field must everywhere be away from B. In the longer part Bm, the net field varies monotonically, at B is large and towards M, and next to m is large and towards m. It will have the right value somewhere in between. Therefore L1 exists in Bm. If the masses are equal, B and L1 are in the centre.
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The method of a A Discovery, 2011-07-23 proves that L4 and L5 exist at the equilateral points — I would like to have a general argument, like those for L1 L2 L3, to show that they must exist somewhere or there — or, if there is no angular momentum, on an equilateral ring.
To Be Merged
The above arguments for L1 to L5 hold whatever the motions of the bodies may be – hyperbolic, parabolic, elliptical, circular, or linear.
In particular, if particles are placed at the L-points, and they and the bodies are released with zero velocity, the system will collapse uniformly into the barycentre. And if they are all released with appropriate infinitesimal velocities, they will reverse direction on reaching the barycentre, describing an infinitely narrow orbit.
To Be Merged
Lagrange actually considered three bodies each of appreciable mass, finding that any three bodies situated in a straight line or an equilateral triangle can retain their pattern.
For a system of three or more masses affected only by mutual forces, consider how their pattern can remain constant, disregarding variations of overall position, orientation, or scale. The barycentre ("centre of gravity"), which does not accelerate, may be considered as part of the pattern. Such a pattern may or may not be stable against small perturbations.
Static Barycentre : In the absence of external forces, the velocity of the barycentre will be constant. Motion of the pattern as a whole is uninteresting, and inertial (non-rotating) coordinates centred on the barycentre can be used.
Inverse square force : For inverse-square gravitation, the relative acceleration of any two parts of such a constant pattern must be in constant proportion to the inverse square of the current size of the pattern.
Central net force : The total angular momentum about the barycentre is constant, and must be divided in a constant fashion among the masses. So the angular momentum of each mass around the barycentre is constant, and the net field at each mass will therefore be (bary-)central.
Conic section : Newton has shown that the path of a mass subject only to a central inverse-square net force is a conic section (circle, ellipse, parabola, hyperbola, straight line). Any conic section is a possible path. Laws similar to Kepler's will apply.
Coplanarity : Three masses define a plane, and the forces between them lie within the plane. If the motion of the masses is in the plane, it will therefore remain so. It remains to be shown that the initial conditions can be satisfied, and can only be satisfied, if the initial velocities with respect to the barycentre lie within the plane.
Three masses can remain in a (self-similar) constant pattern in two distinct ways, with no more than the position, the size and the absolute orientation of the pattern varying with time, as shown below. The paths of the masses are similar conic sections sharing a focus. By taking one mass to be negligible, and considering near-circular paths, one obtains the customary points L1 to L5 described in Gravity 4 : The Lagrange Points and many other places.
The sections below follow Lagrange in using the distances between the masses. But the constant-pattern solutions are proved directly, without considering the general three-mass problem. I have not seen this done elsewhere.
An equilateral triangle ABC with opposite sides a b c remains equilateral if the variable rates of change of the lengths (db/dt, etc.) of its sides are always equal. If initially equal, those rates remain equal if their own variable rates of change (d2b/dt2, etc.) are always equal.
For masses A B C at the corners of an equilateral triangle currently of side s, the gravitational field component at A along side b is G(C+B/2)/s2 and that at C along side b is G(A+B/2)/s2. So at any instant d2b/dt2 is G(A+B+C)/s2, etc., so that d2a/dt2 = d2b/dt2 = d2c/dt2, and the triangle remains equilateral.
Note : that applies for any three masses, without considering the shape of their paths, and does not require the inverse square law.
Except in professional works,
such a proof was first presented, I think, in a section of Gravity 4 : The Lagrange Points, now transferred to A Discovery, 2011-07-23.
Let masses A B C be initially in a straight line. Motion of their barycentre can be disregarded. Let their initial velocities be mutually parallel and in proportion to their distances from the barycentre. The masses will initially remain collinear, and the ratios of their mutual distances will initially remain the same. The pattern will initially be constant.
If AB is infinitesimal, and BC is not, the accelerations will be such that AB/BC will clearly reduce; and vice versa. There will be an intermediate initial ratio AB/BC for which AB/BC remains constant. All of the initial conditions then still hold, and the pattern is preserved for ever.
The traditional Lagrange Points L1, L2, L3 correspond to setting the masses A B C to represent in any order a primary mass, a secondary mass, and a particle.
For the velocities to remain proportional to the distances from the special point the accelerations, and hence the fields, must be similarly proportional. For that, one must construct the necessary equations (which reduce to a single quintic equation) and solve them. That is done in detail in section "Solving Collinear Pattern - L1 L2 L3" in The Lagrange Points Reconsidered, which includes the algebra for the quintic and a Form for the iterative solution of the equations.
Note : that applies for any three masses, without considering the shape of their paths.
Similarly, it is shown in Exactly Two Constant-Pattern Solutions Exist that three-body constant-pattern motion requires the configuration to be either collinear or equilateral; there are no other solutions.
A rbomboidal pattern can be preserved. Any coplanar regular-polygon pattern with equal masses at each corner and symmetrical initial velocities will be preserved.
For more detail and further work, see in The Lagrange Points Reconsidered which includes a possible definition and a proof that there are no other solutions.
The material of page The Lagrange Points Reconsidered, which is shown in the frame below, was originally written for a different means of delivery. The section Masses Moving in Constant Patterns above presents some of that material in a manner rather more like that of my other Web pages.