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Newton’s Great...
Oversight
Galileo’s Falling Bodies and
Lagrange’s Trojan Asteroids
With Their Tadpole and Horseshoe Orbits
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1
INTRODUCTION AND HISTORY
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SECTIONS
1.0 The Gravity of
Falling Apples...
1.1 Aristotle
1.2
Galileo
1.3 Newton
1.4
Lagrange
1.5 Einstein and His “Relativity” |
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1.0
The Gravity of Falling Apples...
Newton’s theory of gravity is over 300 years old, so
old and so accepted that it just couldn’t be wrong about
something as basic as the finding of Galileo that lighter and heavier
bodies fall at the same rate, could it?! Well... yes,
it could. Although science is capable of achieving great insights, it is
also capable of terrifyingly great oversights. Even the greatest of
scientists, such as Sir Isaac Newton, have inexplicably made such
oversights in their chosen fields of scientific endeavor. In particular,
Sir Isaac failed to question the famous finding of Galileo — partly
experimental, partly hypothetical-theoretical (extrapolating to the
general case from the experimental) — that lighter and heavier bodies fall at precisely the same rate. If he had, Newton instead of Lagrange
would undoubtedly have been credited as the theoretical discoverer of
Trojan points (see
Figure 4) and Trojan asteroids.
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DEFINITION — FALLING: By “falling” we
of course mean the net acceleration of the point centers of mass toward
each other, in particular of each of the 2 — lighter/heavier — bodies with
respect to e.g. the Earth, Jupiter, the Sun. It is important to note that
this means the falling or acceleration is measured in the reference frame
of one of the masses that is engaged in the falling or acceleration. Newton theorized
(as did Einstein) that all
masses fall toward each other, so, retroactively, Galileo is stuck with
this definition, even if he thought of the Earth as unmoving.
The hypothesis that lighter and heavier bodies
fall at the precisely the same rate has a simple
counter-example that should have been noticed by Newton’s contemporaries
as well as — especially — by Newton himself:
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the apple falls toward the Earth, but...
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the Earth also falls toward the apple
(Newton’s famously great insight into gravity)
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the Earth will fall faster toward a heavier apple than toward a lighter
apple
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therefore the Earth and the heavier apple will fall together faster,
both Newton-theoretically... and actually
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i.e. the heavier apple falls faster (Earth relative) than the lighter
apple
NOTE: when they are released in separate gedanken trials;
the difference between a 1 kg body and a 2 kg body will be that the Earth’s
initial instantaneous acceleration will be ~1.64·10-24
m/s2
faster for the heavier body, a difference that is
~1.67 parts in 1025 of the standard acceleration of 9.8
m/s2 (i.e. the ratio of the 1 kg mass difference of the
2 bodies to the mass of the Earth); this is well beyond present experimental accuracy
of about 1 part in 1011,
which would make it essentially impossible to measure experimentally, at
this time.
NOTE: this should also have been noticed and analyzed thoroughly (and
publicly) by Einstein, since it at first seems to contradict
relativity theory, which requires that lighter
and heavier test particles “accelerate” at the same
rate, as they do with Newton but only “relative” to Newton’s
absolute space-time (frame of reference),
which theoretically cannot exist in relativity. (This
gets sorted out, more or less in Einstein’s favor.)
It should also have been
noticed that when released simultaneously (see
Figure 1):
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the 2 apples fall toward the Earth, but...
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the 2 apples also fall toward each other, and at different rates because
their masses are different
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since they are equidistant from the center of the Earth, they form an
isosceles triangle with the Earth, and
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the falling rates of each apple toward the other are functions of (among
other things) their angular separation (with the Earth as the vertex), and
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because neither of the equal angles if an isosceles triangle can be
a right angle (i.e. 90°), we have that
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the falling rates of each apple toward the other have different
non-zero components in the direction of the center of the Earth, and also
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theoretically (and actually) the Earth falls slightly faster toward the
heavier apple
The
one fascinating exception:
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unless (as we will show later) the 2 apples with the Earth form
that special variant of an isosceles triangle, an equilateral triangle, in
which case they fall to Earth at precisely the same rate —
Newton-theoretically (Lagrange...
oversighted that his Lagrangian points
L4 and L5 have this property of same falling rates, and they are the only
points that do have it, and then only when precisely 3 bodies are involved)
This falling rate difference can be small, to be
sure. To help give you the idea, the acceleration due to gravity at the
surface of the Earth is about 9.8 m/sec2, but the theoretical
falling rate difference of a 1 and a 2 kilogram mass held 1 meter apart
at the surface of the Earth is only about m/sec2.
Although it is most often very small, the falling rate difference of the 2
bodies has a chance to make itself noticed in the prolonged fall of bodies
in orbit. It is essential to the underlying dynamics of Lagrange’s Trojan
points, L4 and L5 (± 60° angular/orbital separation),
and the “tadpole” and “horseshoe” orbits associated with them.
In fact, the falling rate difference of lighter and
heavier bodies is far more noticeable astronomically than the
“infinitesimal” advance in the perihelion of the orbit of Mercury, since
the Trojan asteroids are highly astronomically observable, and can be
studied in far less than a century. In actual fact, many such asteroids
orbit a single Trojan point in “tadpole”
orbits, and some even orbit both Trojan points in “horseshoe”
orbits. (See
Figure 5.) The most numerous found so far are those asteroids that
inhabit the tadpole orbits around Jupiter’s L4 and L5, but others have
been found associated with Jupiter’s moons, and even Mars. Although a
Trojan asteroid might take hundreds of years to complete its tadpole orbit
of e.g. a Jupiter Trojan point, once such an asteroid is detected, it
takes far less than a century to determine that orbit.
By comparison: the observed advance beyond that
predicted by Newtonian theory in the perihelion of the orbit of Mercury,
that has been studied and determined over the last 100+ years, is
~ 40 arc‑sec/century. If we look at the total angle swept by Mercury in a
century, this advance will be on the order of approximately 10-8
of that total. Changes and advances in the technology of astronomical
observations in the last hundred years have been tremendous, and
accumulated errors and inconsistencies, especially from and with early
observations, are very difficult to gauge. But... it is very close to
the value predicted by Einstein, and is taken as support for relativity
over Newtonian theory. So, although small, it is considered very
important.
Before getting to the equations and further
commentary, we will take a quick look at the history of the falling rate
difference contretemps. Be prepared for frequent references to Newton’s,
and other...
oversights.
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SECTIONS
1.0 The Gravity of
Falling Apples...
1.1 Aristotle
1.2
Galileo
1.3 Newton
1.4
Lagrange
1.5 Einstein and His “Relativity” |
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1.1
Aristotle
Aristotle (384-322 BCE) was a fantastically
famous Greek philosopher-scientist (and perhaps he still is), known among
many other things for being the tutor of a young boy named
Alexander, later known as the Great. Aristotle was so influential that,
although a “pagan”, he was solidly adopted into the pantheon and dogma of
the church as an unquestionable authority (almost the only kind there is,
though some are more unquestionable than others; Aristotle was very
high up in that pecking order, all the more because he was dead by that
time). Until quite recently (the last couple hundred years or so) it was
considered serious heresy to question his teachings. Copernicus
(1473-1543) questioned Ptolemy (ca. 100-178), but would not — or perhaps
could not — question Aristotle.
Many people know that, among many other
things, Aristotle held that heavier bodies fall faster than lighter ones.
If he had stopped there he would have been at least partly scientifically
correct since the Earth and the heavier body will fall together faster,
not much faster, but both theoretically and actually faster (if released
in separate trials). But Aristotle also held that a body would achieve its
final velocity at the instant of release, as if “impulses” existed, but
not “fields” and “action at a distance”. I.e. it was as if he thought of
gravity as supplying an instantaneous impulse that acted only as the body
was released, and that after that it continued with no forces acting on
it, in uniform, unaccelerated motion. (Actually, that is not quite
correct: Aristotle held that “forces” would keep the body in motion until
they ceased and the motion also ceased, a different concept of “forces”
and “motion”, to be sure.) This should all sound a little familiar,
but it needs only a very coarse subjective experience of falling bodies
here on Earth to find untrue. (Visitors from other dimensions, however,
could easily find us boringly provincial.)
Amazingly influential even today at the beginning of
the third millennium of the Common Era, Aristotle was still a quasi-deity
1900 years after his death, when a certain Galileo started making trouble,
questioning the established modern science of his time.
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SECTIONS
1.0 The Gravity of
Falling Apples...
1.1 Aristotle
1.2
Galileo
1.3 Newton
1.4
Lagrange
1.5 Einstein and His “Relativity” |
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1.2
Galileo
It isn’t until Galileo Galilei (1564-1642) that we
have well known historical records of anyone not only disputing
Aristotle’s position, but performing experiments to verify that heavier
bodies do not fall faster than lighter ones, rather that they fall
at the same rate — or to be more precise, observably the same rate under
the conditions of Galileo’s experiments. Galileo left records of rolling
different weight balls down an inclined plane, which allows for more
accurate timing than dropping them from the Tower of Pisa (even if it
introduces other considerations such as rolling friction and angular
momentum). He found them to roll at the same rate, and he hypothesized
that they fall at the same rate.
Galileo also left records of a gedanken experiment —
a thought experiment — that asked one to imagine dropping 2 different
weights with a chain attaching them to each other, and then to think that
the chain does not go taut if they are released at the same instant — and
of course to compare this with their own, the readers’, experience of
reality. Then one was to imagine dropping them with no chain connecting
them and to think that they must fall at the same rate, with
the presence or absence of the chain assumed not to affect the falling
rate. (This was also part of the basis of his same falling rate
hypothesis.) The romantic but almost certainly apocryphal story we all
know has Galileo dropping 2 such bodies from the top of the Tower of Pisa.
It is surprising how many people today — especially
scientists, and even physicists and educators — still believe and teach
that Galileo was scientifically correct when he hypothesized that lighter
and heavier bodies fall at (precisely) the same rate. The truth is, as
usual, far more fascinating. This is not to say that Aristotle was
completely correct, that heavier bodies invariably fall faster than
lighter bodies relative to e.g. the Earth, but they do much of the time.
This can be demonstrated mathematically — and astronomically — even when
deliberately “abstracting out”, i.e. neglecting and/or ignoring, other
real world effects such as viscosity (for purposes of simplification, etc,
as is usual in science), i.e. neglecting everything but Newtonian gravity
and mechanics applied to 3 point masses. With regard to falling rate
differences of lighter and heavier bodies, scientists have neglected since
the days of Newton that the approximations involved hold, or are
extrapolateable, only within limits. This is an all too common failing in
science.
The far more interesting failing of Aristotle’s
theory — thesis might be better; no hypothesis for this guy — is
one shared with Galileo’s. If the 2 bodies are released at the same
instant, the lighter one accelerates toward the heavier one faster than
the heavier one accelerates toward the lighter one, and both these
accelerations have vector components in the direction of the center of
mass of the Earth. Paradoxically, whenever they are closer together than
60 degrees, as they would be if Galileo had actually dropped them from the
Tower of Pisa, the lighter body falls faster! Ironically, Aristotle
and Galileo were both wrong!
The falling rate difference, although usually
very small, is perhaps directly detectable near the Earth’s surface
by today’s laser equipment, but in any case it is readily detectable
astronomically in certain cases of orbiting bodies. The prolonged orbital
fall gives them enough time for the falling rate difference to affect
their orbital positions in a way visible to even the telescopes of the
1800s (well, 1906), when astronomers finally discovered the “Trojan
Planets” Lagrange had predicted over a hundred years earlier. (Or was it
1904? See
http://cfa-www.harvard.edu/cfa/ps/pressinfo/TheFirstTrojanObs.html for
more of the fascinating history of the discovery of the Trojan asteroids.)
One of Galileo’s main concerns was to refute
Aristotle, who had established as scientific dogma, among other things,
that heavier bodies fall faster than lighter ones. This kind of thing —
refutation — is all too often considered dangerous, and Galileo almost
lost his life — tenure of sorts — to the Inquisition for questioning the
still accepted modern scientific dogma of his day. (Remember: Ptolemy had
not yet been traded in for Copernicus by the Keepers of the Flame of
Modern Science, who were about to be traded in themselves.) For example,
Galileo’s Dialogue Concerning the Two Chief Systems of the World,
published in 1632, was put on the Index by the Church, where it remained
until 1822.
(By way of ecumenism: a few years ago — in 1993 —
after a special Vatican commission finished its investigation of the
matter, Pope John Paul II issued a reassessment of the famous 1633 case.
He said that Galileo Galilei was unjustly condemned by the Roman Catholic
Church for promoting a Copernican cosmology.)
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SECTIONS
1.0
The Gravity of Falling Apples...
1.1
Aristotle
1.2
Galileo
1.3 Newton
1.4
Lagrange
1.5 Einstein and His
“Relativity” |
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1.3 Newton
By the late 1600s, Sir Isaac Newton (1642-1727) had
formulated his still famous theory concerning gravity. He may have had
immense help — e.g. in arriving at the inverse square law of gravity —
from Robert Hooke (1635-1702), but Newton both outlived him and
outshone him. Newton’s great insight
— or rather, among the many great
insights attributed to him — was the concept that the same gravity that
attracted the apple to the Earth also attracted the Earth to the apple,
and even 2 apples to each other.
Although he made it explicit in his theory that the
(mass of the) Earth was gravitationally attracted to any other
mass, and that the Earth fell through space as did all other masses, he
slipped up scientifically with regard to an important consequence of his
theory. It is almost totally inexplicable, but Sir Isaac did not use his
theory to carefully analyze the well-known, paradigmatic (in the sense of
Thomas Kuhn), no-falling-rate-difference hypothesis of Galileo. It is also
difficult to understand that neither did his lesser contemporaries — nor
have any physicists or astronomers from their day to the end of the 20th
Century — since it turns out that only simple algebra and trigonometry are
needed to show not only a non-zero Newton-theoretical falling rate
difference, but to (begin to) predict the possible existence of Trojan
asteroids orbiting the Lagrangian points L4 and L5.
Galileo’s falling body problem is essentially a
tractable example, even a very simple one, of the classically intractable
n‑body problem for inverse square fields (n 8 3),
and this has been ignored since the time of Newton’s Principia.
Newton’s Principia (Philosophiae naturalis principia mathematica)
was only first published in 1687, at the insistence of the English
astronomer Halley who also paid for the printing, though Newton had
conceived his theory of gravity many years earlier. Newton’s Principia
was eventually very influential.
With everything else he did, it is excessively
strange that Newton did not question Galileo’s finding, so much so
that it bears much repetition. If he had, he certainly would have
noticed that it was scientifically incorrect, especially in the context of
celestial mechanics. After all, it was Newton who had the brilliant
insight that any mass exerts a gravitational force on any other
mass. He, at least, should have noticed the asymmetry of the masses of
Galileo’s 2 falling bodies and guessed that the falling rate difference
must not only be non-zero, but noticeable if the bodies were in orbit. If
he had, the credit for the theoretical discovery of the Trojan points and
the planets or asteroids which potentially inhabit them would almost
certainly have gone to him, and not to Lagrange.
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SECTIONS
1.0
The Gravity of Falling Apples...
1.1
Aristotle
1.2
Galileo
1.3 Newton
1.4
Lagrange
1.5 Einstein and His
“Relativity” |
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1.4
Lagrange and His
“Trojan Planets”
Joseph Louis Lagrange (1736-1813) was a famous
Italian or French mathematician. (It depends on who you ask. He was born
and baptized Giuseppe Lodovico Lagrangia in Turin, Italy, where he lived
till he was 27, and became a well-known mathematician. He then went to
France for almost 2 years, after which he lived mostly in Germany and,
later, France, again. He perhaps did most of his important mathematics in
France, with Germany a close second.)
Among many other important accomplishments, Lagrange
is credited with developing perturbation theory to solve special
subclasses of the generalized, gravitational n‑body problem for
masses in an inverse square field. Perturbation theory is not considered
simple even by today’s standards, even for professional physicists and
astronomers. (It starts with differential equations, considered essential
in physics but difficult to master, and makes them look simple by
comparison.)
It is important to NOTE that, on the surface to the
uninitiated, Lagrange seems to be solving a 3-body problem, but since he
considered 1 of the 3 bodies to be “infinitesimal”, he was actually
solving a special 2-body problem, or perhaps a fractal 2+body problem.
Well before Lagrange it was known that 2 spherical bodies follow conic
sections with their common center of mass at a focus, and that also with
respect to the other (as the origin in a coordinate system) each will
follow a conic section. Kepler’s laws of planetary motion (1609) had
placed the Sun at the focus of the elliptical orbits of the planets, but
later that century, Newton’s law of gravity (1687) showed that it must be
instead the center of mass of the 2 bodies at the focus of the conic
section. Using his perturbation theory, Lagrange added a ghostly 3rd,
“infinitesimal”
body to the picture, with some inspiring success.
In 1772 Lagrange published a memoir predicting the
potential existence of what he called “Trojan Planets” in the orbit of
Jupiter, but leading/following Jupiter by ± 60°. In fact Lagrange found
that all “homographic
solutions” (see definition in the
APPENDIX) for 3 non-collinear bodies are equilateral triangles, and
that together with the 3 distinct collinear configurations that he also
found, they comprise all homographic solutions for 3 bodies. His theory
also predicted that such a system (i.e. the 3 bodies in an equilateral
triangle) will remain in a “stable equilibrium”, within limits, even when
“perturbed” by other forces such as the gravitational effect of other
planets (thus the name “perturbation theory”). It must be noted, however,
that Lagrange’s concept of “stable equilibrium” allows the 3 bodies to
diverge quite greatly from an equilateral triangle. Modern astronomy’s
concept allows even more. This allowed divergence is today so great that
both
“tadpole” and “horseshoe” orbits (see
Figure 5) are considered regions of that stability.
It was a limitation of Lagrange’s theory, however,
that 1 of the 3 bodies had to be effectively “infinitesimal”
(more on which later). In Lagrange’s day, if one did not make this
simplifying assumption, then the problem reverted to the usual
intractability of the more general 3‑body problem. Lagrange also showed
(within the limits of his assumptions) that, for the Trojan points (i.e.
L4 and L5), 1 of the 2 non-infinitesimal bodies had to be much
smaller than the other (for “stability”; more on this later). This
limitation is no longer necessary computationally, and it is certainly not
desirable theoretically (although advancing theory to take it into account
may still prove quite difficult). The cumulative effect over time of the actual
“non-infinitesimality” of the third mass may lead to a substantial
divergence of theory and reality.
More than a century after Lagrange, in February 1906 (with
some possibility of historical error), the astronomer Max Wolf finally
proved that Lagrange was correct 134 years earlier by
discovering the first Trojan asteroid, 588 Achilles, in the leading
point of Jupiter. His discovery is also the first historically verifiable scientific observation
that corroborates the falling rate difference of lighter and heavier
bodies predicted by Newton’s laws.
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SECTIONS
1.0
The Gravity of Falling Apples...
1.1
Aristotle
1.2
Galileo
1.3 Newton
1.4
Lagrange 1.5 Einstein and
His “Relativity” |
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1.5
Einstein and His
“Relativity”
Newton and Einstein are the 2 biggest names in modern
(Western) science. Even if
we say that Newton was-is its King Arthur (or
even greater),
Even with his sour personality, Newton achieved recognition and acclaim
by the end of his life, and
worship after it. But Einstein gained more world
recognition, not only for himself, but for science, and perhaps more
importantly for the possibility
— even the necessity
— of combining it with a warmly
compassionate humanism.
Albert Einstein means physics to most people today in a way that Newton may
never do (again?). People still find themselves infatuated with Einstein the
person, Einstein the human being, in a way that they never have felt about
Newton.
This is a lead in to mentioning that there is a
question: if lighter and heavier bodies “fall” at different rates,
especially in the case when they are released in separate trials, what
will this mean for Einstein’s theory since relativity requires that
lighter and heavier “test particles” always “accelerate” at the same
rate. It all hinges on the difference between “fall” (“relative” to
other bodies) and “accelerate” (“relative” to an “absolute” space-time
frame of reference). Relativity requires the gedanken existence of
Lorentz frames of reference, i.e. of frames of reference that move in
“uniform” or “unaccelerated” motion, so that the acceleration does not
induce forces that mimic gravity. (Embarrassingly, this uniform-unaccelerated
motion must be with reference to an “absolute” Newtonian-style frame of
reference.) Although Earth is a Lorentz frame to a good approximation
for many classical purposes, this approximation does not extend to the
difference in falling rates of lighter and heavier bodies in the sense
of Galileo and Newton. In a “gedanken real world situation”, lighter and
heavier test particles will accelerate at the same rate with respect to
a frame of reference that is not accelerating with respect to a
Newtonian-style absolute frame of reference at the precise instant of
release (t = 0) in separate trials, but thereafter (t > 0, even
“infinitesimally greater) will accelerate at different rates because
they have caused all the other bodies/masses to accelerate differently,
yielding a different gravitational potential in which they are
accelerating.
In fact, Earth is not really a Lorentz/inertial frame
of reference, and that, even though it implies the existence of an
absolute Newtonian-style frame of reference, is needed for Einstein’s
test particles.
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