© J R Stockton, ≥ 2014-01-25

Links within this site :-

**Merlyn Home Page**- Site Index, E-Mail, Copying- Astronomy / Astronautics 1
- Astronomy / Astronautics 2
- Astronomy / Astronautics 3
- Astronomy / Astronautics 4
- Astronomy / Astronautics 5
- Gravity 0 : Gravity Index :-
- Gravity 1 : Newtonian Gravity
- Gravity 2 : Kepler, etc.
- Gravity 3 : Orbits, Falling, Miscellaneous
- Gravity 4 : The Lagrange Points
- Gravity 5 : A Lagrange Movie
**This Page**:-- History
- General Arguments (to be merged)
- Masses Moving in Constant Patterns
- Framed - The Lagrange Points Reconsidered

- The Lagrange Points Reconsidered
- The History of the Lagrange Points
- Gravity 8 : Hill, Horseshoe, Tide, Roche
- Translations :-

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.

Common explanations of the Lagrange Points are unsophisticated.

Accounts of what Lagrange actually wrote are commonly incorrect.

Accounts of what Lagrange actually wrote are commonly incorrect.

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.

Saturday 2014-01-25, The Daily Telegraph, Weekend p.W17,
"Pubquiz" (Gavin Fuller) :

**Snorter of the week:**

The five points in an orbital configuration where a small mass can orbit in a constant pattern with two larger masses are known as what?"

Answer given : "Lagrange points"

The five points in an orbital configuration where a small mass can orbit in a constant pattern with two larger masses are known as what?"

Answer given : "Lagrange points"

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

Lagrange considered three massive bodies, and no particles.

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.

>To Be Merged

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.

To Be Merged

Consider
the release of a particle in the plane with a velocity appropriate for
its position in the pattern. If it is released in the area near to M or
the area near to m, it will surely move from its place in the pattern
towards that mass. Those areas must meet at L1. If it is released in the
area far from M and m, it will clearly escape to infinity. That area
must meet the inner areas at line borders, and those lines, if not
tangent at L1, must meet at points on each side of the line between the
bodies, probably a moderate distance out. Therefore, it is probable that
L4 and L5 exist.

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.

Fewer
than three masses will necessarily form a constant and stable pattern,
however they may move, except perhaps when two masses collide.

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.

**Initially :**Their initial velocities are such that the ratios of their mutual distances are initially preserved.**Continuously :**Their accelerations are such that the relationship between their velocities and their distances is preserved.

**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
(*d ^{2}b/dt^{2}*, 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)/s ^{2}*
and that at

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 "Collinear Pattern - L1 L2 L3" at 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.

- Introductory : Gravity 4 : The Lagrange Points
- External reference lists are at :-

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.