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The Lagrange Points Revisited
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Dr J R Stockton
Feb 27, 2012, 2:45:18 AM2/27/12
to
From the point of view of almost any person who is not a professional
mathematician or orbital dynamicist, but who has some interest in the
Solar System and who knows as much maths as one starting a science
degree should have or should soon get, I believe that :
(1) It is commonly thought that, in the Earth-Moon system (and others
likewise), there are five Lagrange points at which a particle is more-
or-less stable, revolving about the Earth at the same rate as the Moon
does. L1 is between the Earth and the Moon, nearer to the Moon; L2 is
behind the Moon; L3 is in the other direction; L4 and L5 are in the
Moon's orbit, leading or lagging by 60 degrees. That's about right,
too, but is not completely accurate.
(2) Some are aware, or would be if they thought about it, that the
positions of L1 L2 L3 can be easily enough found, for circular orbits,
by balancing gravitational and "centrifugal" forces; they may even know
that each case gives a different quintic equation, numerically soluble.
(3) They may also have seen a diagram for L4/L5, showing (with some
algebra) that the combination of the fields of the Earth and the Moon at
the particle is as needed.
(4) Some say that Lagrange used such methods. It has been said that
Euler did L1 L2 L3 before Lagrange. At least one says that Lagrange
used Lagrange's Undetermined Multipliers (that may be so, but he did not
actually call them that).
That's what I think is generally thought, from my reading of a few books
and a number of Web sites, and disregarding the writings of ordinary
journalists as being at best worthless.
Please, comments about "what Lagrange wrote" only from those who
have at least seen the text of his "Essay".
Now some Real Truth. All work referred to can be found via links in
<http://www.merlyn.demon.co.uk/gravity4.htm#Refs> and
<http://www.merlyn.demon.co.uk/lagrpapr.htm> which is also framed in
<http://www.merlyn.demon.co.uk/gravity6.htm>.
Therefore, I decided to read what Lagrange wrote. Evidently few moderns
have done so (it is encoded in French). As my French is not good enough
for me to both read and fully understand the original at the same time,
I made a translation, since checked and polished with the aid of a
kindly Frenchman - it is in essai-3c.htm, on my site.
As Euler's work seemed related, I likewise translated, with assistance,
some of Euler's Latin, : euler327.htm, on my site. Euler wrote three
related papers; the first is E.327, the later ones seem to add nothing
relevant. He argued that before tackling the general three-body problem
one must first solve the problem of three bodies moving along a fixed
straight line, and apparently found that argument to be so convincing
that he did not try to get any further. As his line is fixed, he is a
red herring -- the three points he found are too special to qualify as a
discovery of what we now think of as L1 L2 L3.
But Lagrange was trying to solve the general three-body problem, which
is now proven to be insoluble by the tools of the time. After some
hundreds of equations, he began a new chapter, in which, starting from
those many equations, he managed to detect that they support two classes
of constant-pattern solutions : L1 L2 L3, and L4 L5.
So I wondered whether one could easily enough make progress by following
Lagrange's approach of concentrating on the distances between the bodies
without bothering much (if at all) with their positions, but avoiding
any consideration of the general 3-body motion.
For L4 L5, it is remarkably easy - a little thought about constant-
pattern solutions, and, for any three masses, the equilateral
configuration is shown to remain so, with appropriate initial
conditions, perpetually. Here it is ...
Put masses A B C at the corners of an equilateral triangle of side d,
with symmetrical initial velocities such that the lengths of the sides
are changing at equal rates. Consider the second derivative of side BC;
it is minus the sum of the components of acceleration of B and C towards
each other. For B, that acceleration is (C + A cos(60)) G/d^2; for C,
(B + A cos(60)) G/d^2; so d^2(BC)/dt^2 = - (C + B + A) G/d^2 -- and the
same similarly for the other two sides. Therefore the sides continue to
change at equal, though varying, rates. The pattern remains constant.
Of course, this is not a discovery of the points - Lagrange did that -
neither does it show anything about the stability against small
perturbations or imperfect initial conditions, which Lagrange did not do
either.
For L1 L2 L3 being any three collinear masses, I considered the
conditions needed for the ratio of the distances to be preserved. This
leads, permitting but not considering a general rotation, to a single
quintic. Clearly, by making one body massive, another light, and the
third a particle, that quintic must give the well-known three distinct
quintics. But the argument also applies whatever the rotation may be,
so covering motion along any type of conic section, including Euler's
fixed straight line.
I've also looked into the period of the motion, getting appropriate
answers compatible with the Kepler/Newton two-body work.
ISTM pedagogically interesting to see how reference to original work has
enabled common knowledge and understanding to be extended, even though
none of the actual results will be other than well-known to professional
orbital dynamicists.
But I was unable to find a refereed or well-edited publication willing
to present this work where it could also be seen on the Web; thus I can
only use newsgroups, my Web site, and Wikipedia. So I have included that
which I have been offering to the professional media as a linked page on
my Web site, at <http://www.merlyn.demon.co.uk/lagrpapr.htm> and
through <http://www.merlyn.demon.co.uk/gravity6.htm>.
Actually, it had been at "lagrpapr" for some while; but, with no links
pointing to it, Google had not found it.
Minor Query - Did Lagrange actually use (or introduce?) Undetermined
Multipliers on page 256 (or elsewhere) in the Essai?
Main Query - noting how it has been possible to establish the "balance"
of L4 L5, for any three masses in any shape of orbit, in a much better
manner than is usual, by using Lagrange's approach -- are any ingenious
improvements possible in other parts of the topic, without using maths
more profound than the average working scientist will know about?
Secondary query - any other suggestions for improvement? For example
I've ignored the possibility of non-planar motion, but there should be
some simple argument to show that it yields nothing of interest.
Also, there are a few added loose ends, which can be revealed by "1"
buttons, in <http://www.merlyn.demon.co.uk/lagrpapr.htm>, and it would
be nice if they could be tied up.
Any other comment welcome, too.
--
(c) John Stockton, nr London, UK. ?@merlyn.demon.co.uk Turnpike v6.05 MIME.
Web <http://www.merlyn.demon.co.uk/> - FAQqish topics, acronyms and links;
Astro stuff via astron-1.htm, gravity0.htm ; quotings.htm, pascal.htm, etc.
mathematician or orbital dynamicist, but who has some interest in the
Solar System and who knows as much maths as one starting a science
degree should have or should soon get, I believe that :
(1) It is commonly thought that, in the Earth-Moon system (and others
likewise), there are five Lagrange points at which a particle is more-
or-less stable, revolving about the Earth at the same rate as the Moon
does. L1 is between the Earth and the Moon, nearer to the Moon; L2 is
behind the Moon; L3 is in the other direction; L4 and L5 are in the
Moon's orbit, leading or lagging by 60 degrees. That's about right,
too, but is not completely accurate.
(2) Some are aware, or would be if they thought about it, that the
positions of L1 L2 L3 can be easily enough found, for circular orbits,
by balancing gravitational and "centrifugal" forces; they may even know
that each case gives a different quintic equation, numerically soluble.
(3) They may also have seen a diagram for L4/L5, showing (with some
algebra) that the combination of the fields of the Earth and the Moon at
the particle is as needed.
(4) Some say that Lagrange used such methods. It has been said that
Euler did L1 L2 L3 before Lagrange. At least one says that Lagrange
used Lagrange's Undetermined Multipliers (that may be so, but he did not
actually call them that).
That's what I think is generally thought, from my reading of a few books
and a number of Web sites, and disregarding the writings of ordinary
journalists as being at best worthless.
Please, comments about "what Lagrange wrote" only from those who
have at least seen the text of his "Essay".
Now some Real Truth. All work referred to can be found via links in
<http://www.merlyn.demon.co.uk/gravity4.htm#Refs> and
<http://www.merlyn.demon.co.uk/lagrpapr.htm> which is also framed in
<http://www.merlyn.demon.co.uk/gravity6.htm>.
Therefore, I decided to read what Lagrange wrote. Evidently few moderns
have done so (it is encoded in French). As my French is not good enough
for me to both read and fully understand the original at the same time,
I made a translation, since checked and polished with the aid of a
kindly Frenchman - it is in essai-3c.htm, on my site.
As Euler's work seemed related, I likewise translated, with assistance,
some of Euler's Latin, : euler327.htm, on my site. Euler wrote three
related papers; the first is E.327, the later ones seem to add nothing
relevant. He argued that before tackling the general three-body problem
one must first solve the problem of three bodies moving along a fixed
straight line, and apparently found that argument to be so convincing
that he did not try to get any further. As his line is fixed, he is a
red herring -- the three points he found are too special to qualify as a
discovery of what we now think of as L1 L2 L3.
But Lagrange was trying to solve the general three-body problem, which
is now proven to be insoluble by the tools of the time. After some
hundreds of equations, he began a new chapter, in which, starting from
those many equations, he managed to detect that they support two classes
of constant-pattern solutions : L1 L2 L3, and L4 L5.
So I wondered whether one could easily enough make progress by following
Lagrange's approach of concentrating on the distances between the bodies
without bothering much (if at all) with their positions, but avoiding
any consideration of the general 3-body motion.
For L4 L5, it is remarkably easy - a little thought about constant-
pattern solutions, and, for any three masses, the equilateral
configuration is shown to remain so, with appropriate initial
conditions, perpetually. Here it is ...
Put masses A B C at the corners of an equilateral triangle of side d,
with symmetrical initial velocities such that the lengths of the sides
are changing at equal rates. Consider the second derivative of side BC;
it is minus the sum of the components of acceleration of B and C towards
each other. For B, that acceleration is (C + A cos(60)) G/d^2; for C,
(B + A cos(60)) G/d^2; so d^2(BC)/dt^2 = - (C + B + A) G/d^2 -- and the
same similarly for the other two sides. Therefore the sides continue to
change at equal, though varying, rates. The pattern remains constant.
Of course, this is not a discovery of the points - Lagrange did that -
neither does it show anything about the stability against small
perturbations or imperfect initial conditions, which Lagrange did not do
either.
For L1 L2 L3 being any three collinear masses, I considered the
conditions needed for the ratio of the distances to be preserved. This
leads, permitting but not considering a general rotation, to a single
quintic. Clearly, by making one body massive, another light, and the
third a particle, that quintic must give the well-known three distinct
quintics. But the argument also applies whatever the rotation may be,
so covering motion along any type of conic section, including Euler's
fixed straight line.
I've also looked into the period of the motion, getting appropriate
answers compatible with the Kepler/Newton two-body work.
ISTM pedagogically interesting to see how reference to original work has
enabled common knowledge and understanding to be extended, even though
none of the actual results will be other than well-known to professional
orbital dynamicists.
But I was unable to find a refereed or well-edited publication willing
to present this work where it could also be seen on the Web; thus I can
only use newsgroups, my Web site, and Wikipedia. So I have included that
which I have been offering to the professional media as a linked page on
my Web site, at <http://www.merlyn.demon.co.uk/lagrpapr.htm> and
through <http://www.merlyn.demon.co.uk/gravity6.htm>.
Actually, it had been at "lagrpapr" for some while; but, with no links
pointing to it, Google had not found it.
Minor Query - Did Lagrange actually use (or introduce?) Undetermined
Multipliers on page 256 (or elsewhere) in the Essai?
Main Query - noting how it has been possible to establish the "balance"
of L4 L5, for any three masses in any shape of orbit, in a much better
manner than is usual, by using Lagrange's approach -- are any ingenious
improvements possible in other parts of the topic, without using maths
more profound than the average working scientist will know about?
Secondary query - any other suggestions for improvement? For example
I've ignored the possibility of non-planar motion, but there should be
some simple argument to show that it yields nothing of interest.
Also, there are a few added loose ends, which can be revealed by "1"
buttons, in <http://www.merlyn.demon.co.uk/lagrpapr.htm>, and it would
be nice if they could be tied up.
Any other comment welcome, too.
--
(c) John Stockton, nr London, UK. ?@merlyn.demon.co.uk Turnpike v6.05 MIME.
Web <http://www.merlyn.demon.co.uk/> - FAQqish topics, acronyms and links;
Astro stuff via astron-1.htm, gravity0.htm ; quotings.htm, pascal.htm, etc.
Jonathan Thornburg [remove -animal to reply]
Feb 27, 2012, 6:43:37 AM2/27/12
to
Dr J R Stockton <repl...@merlyn.demon.co.uk> wrote:
> For L4 L5, it is remarkably easy - a little thought about constant-
> pattern solutions, and, for any three masses, the equilateral
> configuration is shown to remain so, with appropriate initial
> conditions, perpetually. Here it is ...
>
> Put masses A B C at the corners of an equilateral triangle of side d,
> with symmetrical initial velocities such that the lengths of the sides
> are changing at equal rates. Consider the second derivative of side BC;
> it is minus the sum of the components of acceleration of B and C towards
> each other. For B, that acceleration is (C + A cos(60)) G/d^2; for C,
> (B + A cos(60)) G/d^2; so d^2(BC)/dt^2 = - (C + B + A) G/d^2 -- and the
> same similarly for the other two sides. Therefore the sides continue to
> change at equal, though varying, rates. The pattern remains constant.
>
> Of course, this is not a discovery of the points - Lagrange did that -
> neither does it show anything about the stability against small
> perturbations or imperfect initial conditions, which Lagrange did not do
> either.
If we simplify to the circular restricted 3-body problem
> For L4 L5, it is remarkably easy - a little thought about constant-
> pattern solutions, and, for any three masses, the equilateral
> configuration is shown to remain so, with appropriate initial
> conditions, perpetually. Here it is ...
>
> Put masses A B C at the corners of an equilateral triangle of side d,
> with symmetrical initial velocities such that the lengths of the sides
> are changing at equal rates. Consider the second derivative of side BC;
> it is minus the sum of the components of acceleration of B and C towards
> each other. For B, that acceleration is (C + A cos(60)) G/d^2; for C,
> (B + A cos(60)) G/d^2; so d^2(BC)/dt^2 = - (C + B + A) G/d^2 -- and the
> same similarly for the other two sides. Therefore the sides continue to
> change at equal, though varying, rates. The pattern remains constant.
>
> Of course, this is not a discovery of the points - Lagrange did that -
> neither does it show anything about the stability against small
> perturbations or imperfect initial conditions, which Lagrange did not do
> either.
(M1, M2 < M1, and a test mass), then the L4 and L5 positions are only
stable if M2/M1 < about 1/25. For less-extreme mass ratios L4 and
L5 are UNstable. (I don't know the status of L1, L2, and L3 offhand,
and I'm currently several time zones away from my reference books.)
Thus the Earth-Moon L4 and L5 points are stable, but many (most)
binary-star L4 and L5 points are unstable.
There's a Hal Clement science-fiction short story in which a character's
lack of knowledge of this fact leads to his death; alas I don't recall
the title.
--
-- "Jonathan Thornburg [remove -animal to reply]" <jth...@astro.indiana-zebra.edu>
Dept of Astronomy & IUCSS, Indiana University, Bloomington, Indiana, USA
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam
Dr J R Stockton
Feb 29, 2012, 1:55:29 AM2/29/12
to
In sci.astro.research message <mt2.0-12947...@hydra.herts.ac.uk>
, Mon, 27 Feb 2012 11:43:37, "Jonathan Thornburg [remove -animal to
reply]" <jth...@astro.indiana-zebra.edu> posted:
>Dr J R Stockton <repl...@merlyn.demon.co.uk> wrote:
>> For L4 L5, it is remarkably easy - a little thought about constant-
>> pattern solutions, and, for any three masses, the equilateral
>> configuration is shown to remain so, with appropriate initial
>> conditions, perpetually. Here it is ...
>>
>> Put masses A B C at the corners of an equilateral triangle of side d,
>> with symmetrical initial velocities such that the lengths of the sides
>> are changing at equal rates. Consider the second derivative of side BC;
>> it is minus the sum of the components of acceleration of B and C towards
>> each other. For B, that acceleration is (C + A cos(60)) G/d^2; for C,
>> (B + A cos(60)) G/d^2; so d^2(BC)/dt^2 = - (C + B + A) G/d^2 -- and the
>> same similarly for the other two sides. Therefore the sides continue to
>> change at equal, though varying, rates. The pattern remains constant.
>>
>> Of course, this is not a discovery of the points - Lagrange did that -
>> neither does it show anything about the stability against small
>> perturbations or imperfect initial conditions, which Lagrange did not do
>> either.
>
>If we simplify to the circular restricted 3-body problem
>(M1, M2 < M1, and a test mass), then the L4 and L5 positions are only
>stable if M2/M1 < about 1/25. For less-extreme mass ratios L4 and
>L5 are UNstable. (I don't know the status of L1, L2, and L3 offhand,
>and I'm currently several time zones away from my reference books.)
>Thus the Earth-Moon L4 and L5 points are stable, but many (most)
>binary-star L4 and L5 points are unstable.
Follow the links I gave and you'll be able to read a lot about the
Lagrange points, including links. I doubt whether what I've written
about in the previous article is in your books - I hope not, anyway.
L1 L2 L3 are always unstable. L4 and L5 are stable, for circular
orbits, providing that one mass is at least 24.9599 times heavier than
the sum of the other two. Writing "sum" there is inexact, but only
slightly so. AFAIK, "sum" is not well-known; but it is easily shown
from the well-established relation
27*(m_1*m_2 + m_2*m_3 + m_3*m_1) < (m_1 + m_2 + m_3)^2
>There's a Hal Clement science-fiction short story in which a character's
>lack of knowledge of this fact leads to his death; alas I don't recall
>the title.
Since reading "Mission of Gravity" I've bought a copy of every book of
his that I've seen on sale, and probably read every short story that
I've otherwise seen. Alas, only a baker's dozen of books. I don't
recall that one, though.
--
(c) John Stockton, near London. *@merlyn.demon.co.uk/?.?.Stockton@physics.org
Web <http://www.merlyn.demon.co.uk/> - FAQish topics, acronyms, and links.
, Mon, 27 Feb 2012 11:43:37, "Jonathan Thornburg [remove -animal to
reply]" <jth...@astro.indiana-zebra.edu> posted:
>Dr J R Stockton <repl...@merlyn.demon.co.uk> wrote:
>> For L4 L5, it is remarkably easy - a little thought about constant-
>> pattern solutions, and, for any three masses, the equilateral
>> configuration is shown to remain so, with appropriate initial
>> conditions, perpetually. Here it is ...
>>
>> Put masses A B C at the corners of an equilateral triangle of side d,
>> with symmetrical initial velocities such that the lengths of the sides
>> are changing at equal rates. Consider the second derivative of side BC;
>> it is minus the sum of the components of acceleration of B and C towards
>> each other. For B, that acceleration is (C + A cos(60)) G/d^2; for C,
>> (B + A cos(60)) G/d^2; so d^2(BC)/dt^2 = - (C + B + A) G/d^2 -- and the
>> same similarly for the other two sides. Therefore the sides continue to
>> change at equal, though varying, rates. The pattern remains constant.
>>
>> Of course, this is not a discovery of the points - Lagrange did that -
>> neither does it show anything about the stability against small
>> perturbations or imperfect initial conditions, which Lagrange did not do
>> either.
>
>If we simplify to the circular restricted 3-body problem
>(M1, M2 < M1, and a test mass), then the L4 and L5 positions are only
>stable if M2/M1 < about 1/25. For less-extreme mass ratios L4 and
>L5 are UNstable. (I don't know the status of L1, L2, and L3 offhand,
>and I'm currently several time zones away from my reference books.)
>Thus the Earth-Moon L4 and L5 points are stable, but many (most)
>binary-star L4 and L5 points are unstable.
Lagrange points, including links. I doubt whether what I've written
about in the previous article is in your books - I hope not, anyway.
L1 L2 L3 are always unstable. L4 and L5 are stable, for circular
orbits, providing that one mass is at least 24.9599 times heavier than
the sum of the other two. Writing "sum" there is inexact, but only
slightly so. AFAIK, "sum" is not well-known; but it is easily shown
from the well-established relation
27*(m_1*m_2 + m_2*m_3 + m_3*m_1) < (m_1 + m_2 + m_3)^2
>There's a Hal Clement science-fiction short story in which a character's
>lack of knowledge of this fact leads to his death; alas I don't recall
>the title.
his that I've seen on sale, and probably read every short story that
I've otherwise seen. Alas, only a baker's dozen of books. I don't
recall that one, though.
--
(c) John Stockton, near London. *@merlyn.demon.co.uk/?.?.Stockton@physics.org
Web <http://www.merlyn.demon.co.uk/> - FAQish topics, acronyms, and links.
Dr J R Stockton
Oct 9, 2012, 2:16:43 AM10/9/12
to
In sci.astro.research message <mt2.0-10837...@hydra.herts.ac.uk>
, Mon, 27 Feb 2012 07:45:18, Dr J R Stockton
<repl...@merlyn.demon.co.uk> posted:
UPDATE :
> It has been said that
>Euler did L1 L2 L3 before Lagrange.
Euler did do L1 & L2 before Lagrange, in E.304.
See <http://www.merlyn.demon.co.uk/euler304.htm> (2012).
>As Euler's work seemed related, I likewise translated, with assistance,
>some of Euler's Latin, : euler327.htm, on my site. Euler wrote three
>related papers; the first is E.327,
But E.304, while not much related, is relevant to L1 & L2.
> the later ones seem to add nothing
>relevant.
QUESTIONS :
Did Euler write anything amounting to a discovery of L3?
Did Euler write anything amounting to a discovery of L4/5?
For those, a link to images of original or other ancient publications is
needed; a link to contemporary HTML cannot be convincing.
All of Euler's publications should be available at the Euler Archive,
<http://eulerarchive.maa.org/">. At least some of his correspondence is
also there.
--
(c) John Stockton, nr London, UK. Mail via homepage. Turnpike v6.05 MIME.
, Mon, 27 Feb 2012 07:45:18, Dr J R Stockton
<repl...@merlyn.demon.co.uk> posted:
UPDATE :
> It has been said that
>Euler did L1 L2 L3 before Lagrange.
See <http://www.merlyn.demon.co.uk/euler304.htm> (2012).
>As Euler's work seemed related, I likewise translated, with assistance,
>some of Euler's Latin, : euler327.htm, on my site. Euler wrote three
>related papers; the first is E.327,
> the later ones seem to add nothing
>relevant.
Did Euler write anything amounting to a discovery of L3?
Did Euler write anything amounting to a discovery of L4/5?
For those, a link to images of original or other ancient publications is
needed; a link to contemporary HTML cannot be convincing.
All of Euler's publications should be available at the Euler Archive,
<http://eulerarchive.maa.org/">. At least some of his correspondence is
also there.
--
(c) John Stockton, nr London, UK. Mail via homepage. Turnpike v6.05 MIME.
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