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Chapter 1

The Laws of Dynamics

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1.1 Kepler’s Empirical Laws

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In order to show the importance of the three-body problem for the emergence of the
idea of chaos, we begin by looking at the set of conceptual problems surrounding
celestial mechanics at the time when Nicholas Kopernik (1473–1543) published his
De Revolutionibus in 1543. He had the magnificent idea to place the sun near the
center of the planetary orbits, following in the footsteps of his Greek predecessors
Philolaus the Pythagorian (5th century BC), who argued that the earth revolved
around a central fire, or Aristarchus of Samos (-310: -230). In spite of this Kopernik
assumed that the only role that the Sun played was to provide light for the Earth.
As did his predecessors, Kopernik was content simply to describe the motions of
the planets, without any real understanding of the causes. Johannes Kepler (1571–
1630) was one of the first to claim that the Sun was responsible for the movement
of the planets. In fact, in his Mysterium Cosmographicum of 1596, he attributed a
real dynamical role to the sun by arguing that it was the source of an attracting
force on all planets, and that this force decreased with increasing distance of the
planets from the sun.
In spite of this insight, Kepler’s name is forever associated with the description
of planetary motion in terms of elliptical orbits. Up to then, planetary orbits
had always been described by uniform circular motions or combinations of uniform
circular motions as had been taught by Aristotle (-384: -322), following Plato’s lead
(-428: -348). The idea that circular motion was fundamental to the description of
celestial motion goes back at least as far as Plato’s Timeus. Aristotle justified
this by stating that circular motion was the only perfect type of motion that stars
could design to follow. It was only after failing to describe planetary motion in this
way over a period of five long years that Kepler became convinced that planetary
motions could only be described by ellipses. He once wrote:1

My first mistake was to assume that the planetary orbits were perfect
circles. This error cost me so much time because it had been propagated
1 J. Kepler, Astronomia Nova (1609), Chap. xl.

3
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4 Chaos in nature

by the authority of all the philosophers, and further was metaphysically

very reasonable.

Aware of the criticism that his contradiction of Aristotelian physics would incite,
Kepler divided his Astronomia Nova (1609) into seventy chapters where he at-
tempted to show that any other description of planetary motion contradicted the
very precise measurements of Tycho Brahe (1546–1601), in which he had absolute
confidence.
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Already persuaded of the dynamical role played by the sun, Kepler naturally
placed the sun at one of the foci of an ellipse (Figure 1.1). For Kepler, the elliptic
shape of planetary orbits was explained by two different contributions. The first
one corresponds to a force emanating from the sun and influencing each planet.
Propagated within the ecliptic plane — the mean plane in which planets’ orbits
were inscribed — this force depends on the inverse of the distance between the sun
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and each planet. For Kepler, this force shall induce a circular motion around the
sun. A second — magnetic — force emanating from each planet was responsible for
varying the distance between each planet and the sun. In other words, the magnetic
force was responsible for the ellipticity and only emanates from each planet, just
because ellipticity was planet dependent. Despite the imperfections of his system,
this was a description of Nature that reduced the role of God to that of a creator:
having given us the laws according to which the world evolves, He no longer needs
to intervene. This view is presented in a letter dated 10 February 1605 as follows.2

My aim is to show that the heavenly machine is not a kind of di-

vine, living being, but a kind of clockwork [...], insofar as nearly all the
manifold motions are caused by a most simple, magnetic, and material
force, just as all motions of the clock are caused by a simple weight.
And I show how these physical causes are to be given numerical and
geometrical expression.

P1
11111111
00000000
11111111
00000000
Earth
11111111
00000000
11111111
00000000 11111111111111111111111
00000000000000000000000
P4
11111111111111111111111
11111111
00000000 00000000000000000000000
11111111111111111111111
perihelion 11111111
00000000 00000000000000000000000
11111111111111111111111
00000000000000000000000 aphelion
11111111
00000000 11111111111111111111111
00000000000000000000000P 3
11111111Sun
00000000
11111111
00000000
11111111
00000000
P2

Figure 1.1 Kepler’s system. Elliptical trajectory of the Earth around the Sun, which is placed at
one of the foci of the ellipse. The point on the trajectory closest to the Sun is called the perihelion
and the furthest point is called the aphelion. The variation in the distance between the Earth and
the Sun was driven by a magnetic force emanating from the Earth.

Kepler believed that the motions of the planets were completely determined,
2 J. Kepler, Letter dated on February 10, 1605, quoted in I. Peterson, Newton’s clock: Chaos in

the Solar system, Freeman, 1993.

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The Laws of Dynamics 5

governed by unchanging laws and evolving like clockwork. He succeeded in describ-

ing planetary motion, within the limits of Tycho Brahe’s observations, by three laws
that were stated as follows:

• the path of the planets about the sun are elliptical in shape, with the center
of the sun being located at one focus;
• an imaginary line drawn from the center of the sun to the center of the
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planet will sweep out equal areas in equal intervals; areas delimited by
P 
1 P2 and P3 P4 are described in equal intervals; thus, the planet moves at
the perihelion faster than at the aphelion (Figure 1.1);
• the ratio of the squares of the mean motions3 n is equal to the inverse of
the ratio of the cubes of the semi-major axe a of their ellipses, that is,
the product n2 a3 = is constant. This law expresses the fact that, the more
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distant from the Sun the planet is, the slower its motion is. It was published
in Kepler’s book Harmonices Mundi (1619).

Despite this, if Kepler obtained very good agreement with the observations of plan-
etary motion, and if his Tables Rudolphines were incontestably the best of the time,
he had more difficulty with the problem of the moon’s motions, which is particularly
sensitive to the motions of the other planets:4

how adding or substracting the “small” lunar orbit, which could not
extend very much beyond the thinness of the Earth’s orbit by very much,
could influence the increase or decrease of all the other spheres.

He stated, as had Kopernik before him, that he was often faced with large deviations
caused by small errors in either the planetary masses or the diameters of their
orbits:5

we are constrained to determine large orbital changes from very small

and almost imperceptible [elements], and from which, due to a departure
by a few minutes, this is 5 or 6 degrees which are lost, and thus a small
error can be immensely propagated.

This rapid growth of small errors in the description of the trajectories of heavenly
bodies is a phenomenon that we run into throughout this work.

3 The mean motion is defined as the mean speed the planet would have if its motion was uniform
circular.
4 J. Kepler, Le Secret du Monde (1596), Transl. A. Segonds, Gallimard, 1993, p. 150.
5 N. Kopernik, Des Révolutions des orbes célestes, Book iii, Chap. 20 quoted by J. Kepler, Le

secret du Monde (1596), Ibid.

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6 Chaos in nature

1.2 The Law of Gravitation

Kepler’s elliptical trajectories involve only two bodies, the more massive being
placed at a focus of the ellipse. Combinations of the ellipse with additional motions
are used to take into account the influence of other bodies. Before defining the
three-body problem we should describe the interactions among several bodies. The
law of gravitation is expressed in this way.
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Towards the middle of the 17th century several scientists attempted to find an
expression according to which the force exerted by the sun decreases as a function
of the distance r which separates the interacting bodies. Kepler imagined a law
depending on the inverse square of the distance as he got for the propagation of
light in his Optica (1604), but he abandoned this idea when he realized the force
seemed to only propagate within the ecliptic plane, that is, according to a circle
perimeter. The force should therefore depend on the inverse law of the distance
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since varying according to a circle and not to a sphere. Jeremiah Horrocks (1617–
1641) was one of the earliest to take up Kepler’s works. While he was a student
at Cambridge University he did this himself because, at that time, there was no
course in astronomy, much less courses describing the works of Kopernik, Tycho
Brahe, or Kepler. Horrocks discovered Kepler’s works by reading the work of Philips
van Lansberge6 in which this Belgian astronomer defended Kopernik against Tycho
Brahe but argued against Kepler’s ellipses. Horrocks convinced himself very quickly
that Kepler’s ellipses were more accurate than uniform circular motion. Horrocks
applied Kepler’s ideas to the interactions between the sun and the planets:7

I, on the contrary, make the planet naturally to be averse from the

Sun and desirous to rest in his own place, caused by a material dullness
naturally opposite to motion and averse to the Sun without either power
or the will to move to the Sun of itself. But then, the Sun by its own rays
attracts and by its circumferential revolution carries about the unwilling
planet, conquering that natural self-rest that is in it; yet not so far, but
that the planet doth much abate and weaken this force of the Sun, as is
largely disputed afore.

Reviewing Kepler’s ideas that the Sun is the source of planetary motion, he proposed
a mathematical law describing the decrease in the force with distance between the
sun and planets. In his theory of the moon, Horrocks observed that although the
Moon is between the Earth and the Sun, the ellipse described by the Moon — with
the Earth at one of its foci — is elongated: this is because the eccentricity and
apogee vary in time. He also studied cometary motion — which he found to be
6 P. van Lansberge, Tabulae motuum caelestium perpetuae: ex omnium temporarum observation-

ibus consentientes, 1632.

7 J. Horrocks, Philosophical exercices, Part 1, para 26, (1661), quoted by Robert Brickel, A Chap-

ter of Romance in Science 1639–1874: In Memoriam Horroccii and as mentioned by V. Barocas,

A country Curate, Quaterly Journal of the Royal Astronomical Society, 12, 179–182, 1971.
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The Laws of Dynamics 7

almost elliptical — and the tides. All these ideas found their way into Newton’s
Principia. In fact, among Horrock’s contemporaries at Cambridge was John Wallis
(1616–1703), whose works were read attentively by Newton, and Ralph Cadworth,
who influenced Newton’s mechanistic philosophy. We point out that Wallis, who
directed Newton to the calculus of changes, was charged by the Royal Society of
London with the publication8 of Horrock’s works in 1666, the year during which
Newton declared that he began his studies of planetary motions. Newton held a
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very high opinion of Horrocks:9

More inequalities in the Moon’s motion have not hitherto been taken
notice of by astronomers. But all these follow from our principles, and
are known really to exist in the heavens. And this may be seen in that
most ingenious, and, if I mistake not, of all the most accurate, hypothesis
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of Mr. Horrox, which Mr. Flamsteed has fitted to the heavens.

and in the first edition of Principia10

Our Countryman Horrox [who] was the first who advanced the theory
of the moon’s moving in an ellipse about the Earth placed at its lower
focus.

Horrocks was an important bridge between the works of Kepler and those of Newton.
Christiaan Huygens (1629–1695) had studied uniform circular motion and es-
tablished that the centrifugal force followed a 1/r2 law. In addition, Robert Hooke
(1635–1703) and Christopher Wren (1632–1723) tried to establish a relation be-
tween the 1/r2 law and the elliptical trajectories of celestial bodies. Specifically,
Hooke developed a qualitative model based on the principle of inertia, correctly
stated by Descartes:11

Each particular part of matter continues always to be in the same

state unless collision with others constrains it to change that state.

He coupled this principle with the equilibrium between the centrifugal force and the
force due to the Sun. If he found motion qualitatively in agreement to observations,
he was not able to obtain results quantitatively in agreement to Kepler’s laws.
Nevertheless, he claimed that the force was not only between the sun and the
planets, as it was in Kepler’s Astronomia, but between any massive bodies:12
8 Horrocks’ Opera were published in 1672.
9 I. Newton, A treatise of the system of the world, London, p. 56, 1728.
10 I. Newton, Principia Mathematica, Book iii, Scholium 475, 1687.
11 R. Descartes, Le Monde : ou Traité de la lumière, in Oeuvres de Descartes, ed. AM-Tannery

(Paris: L. Cerf, 1897–1913), 11, p. 435 (1677 pagination), translated by M. S. Mahoney, New
York, p. 61, 1979.
12 R. Hooke, An attempt to prove the Motion of the Earth from Observations, 1674.
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8 Chaos in nature

all Coelestial Bodies whatsoever, have an attraction or graviting

power towards their own Centers, whereby they attract or not only their
own parts, and keep them from flying from them, as we may observe
the Earth to do, but that they do also attract all the other Coelestial
Bodies that are within the sphere of their activity; and consequently
that not only the Sun and Moon have an influence upon the body and
motion of the Earth, and the Earth upon them, but that also, and by
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their attractive powers, have a considerable influence upon its motion as

in the same manner the corresponding attractive power of the Earth has
a considerable influence upon every one of their motions also.

It was Isaac Newton (1642–1727) who reaped the honor of showing that a 1/r2
force leads directly to Kepler’s laws. To do this he resorted to very subtle geometric
arguments that related motion to infinitely small displacements. He studied dif-
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ferent laws of the form 1/rn and stated, in Book iii of his Principia Mathematica
(1687) that gravity follows a law in which it decreases like the inverse square of the
distances:9

Proposition ii: That the forces by which the primary planets are
continually drawn off from rectilinear motions, and retained in their
proper orbits, tend to the sun; and are reciprocally as the squares of the
distances of the places of those planets from the sun’s centre.

and is universal:

Proposition vii: That there is a power of gravity tending to all bodies

proportional to the several quantities of matter which they contain.

With these two propositions, Newton unified celestial mechanics with the empir-
ical laws of Kepler and the terrestial mechanics of falling bodies studied by Galilei
(1564–1642): he shows that a single force was responsible for all these phenomena.
To show that Kepler’s laws result from a 1/r2 force law Newton used, among other
arguments, the idea of an accelerating force introduced in Book i of the Principia
as follows.

Lemma x: Spaces which a body describes by any finite force urging

it, whether that force is determined and immuable, or is continually
augmented or continually diminished, are in the very beginning of the
motion one to the other in the duplicate ratio of the times.

It was in this form that the fundamental principle of dynamics was presented by
Newton. We had to await the development of differential calculus, independently
discovered by Newton and the “school” of Gottfried Leibniz (1646–1716) for a
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The Laws of Dynamics 9

modern formulation of this principle, stated by Leonard Euler in 1747:13 the prod-
uct of the mass and the acceleration of a body is equal to the sum of the forces on
the body.
Using this fundamental dynamical principle, the geometries, as it was called,
were able to show the relation between changes in motion and the forces that caused
them. Any system in which motion results from applied forces is a dynamical system,
in the sense that its behavior evolves in time. Since this fundamental principle of
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dynamics establishes a relation between an acceleration and the applied forces, it

is expressed by a differential equation which involves the second derivative of the
position — the acceleration — and the position evolving under the influence of the
forces on which it depends.14
In the meantime, the first differential form of Newton’s equation describing the
problem of two massive bodies under gravitational interaction were first written
in 1710 by Jacob Hermann (1678–1733),15 one pupil of Johann Bernoulli (1667–
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1748).16 Both of them opened the door for Euler’s principle. They solved the
differential equations and confirmed Newton’s result that the laws of motion are, in
this case, exactly Kepler’s laws. Thus, when a single planet orbits the sun, it moves
exactly according to the laws found by Kepler. The planet follows an elliptical
trajectory that repeats itself forever: this is a periodic solution. Kepler’s third law
gives us the period of this solution as a function of the distance from the sun as
measured by the semi-major axis a of the ellipse (in astronomical units):

Mercury Venus Earth Mars Jupiter Saturn

a 0.38 0.72 1.00 1.52 5.2 9.5
1/n 88 224 365 687 4307 10767
n2 a 3 7.09 7.44 7.50 7.44 7.58 7.39

where 1/n corresponds to the inverse of the average motion, that is, to the period
of revolution, given in Earth days. Only the planets known to Kepler are presented
in this Table but the near constant value of the product n2 a3 is satisfied for all the
planets (major and minor) known today. As a result, we can consider the two body
problem as solved, and that Kepler’s Laws describe their solutions.
13 L. Euler, Découverte d’un nouveau principe de mécanique, Mémoires de l’Académie des Sciences

de Berlin (1750),6, 185–217, 1752. The modern writing of this principle has the form of a vectorial
equation ma =  where m is the mass of the body, a the acceleration vector and
F F is the
sum of the forces applied on the body.
14 Writing this in a differential form came after the contribution by Pierre Varignon (1654–1722)

who associated the concept of acceleration with the first time derivative of the velocity, a = dv
dt
, or
d2 OM

to the second time derivative of the position, a = dt2
. The fundamental principle of dynamics
2
is written as the differential equation ddt2x = F (x) which is a second-order differential equation
since involving a second time derivative.
15 J. Hermann, Letter to J. Bernoulli, July 12, 1710, in J. Bernoulli, Opera, 85.
16 D. Speiser, The Kepler problem from Newton to Johann Bernoulli, Archive for History of Exact

Sciences, 50, (2), 103–116, 1996.

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10 Chaos in nature

1.3 Theory of the Moon

Newton had already been interested in the problem where three bodies interacted
gravitationally. This configuration occurs when the action of a second body on a
third cannot reasonably be neglected compared to the action of the first body. In
our solar system there are two systems where this type of interaction is particularly
important. These are the Sun-Earth-Moon system, because of the close proximity
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of the Moon to the Earth, and the Sun-Jupiter-Saturn system, due to Jupiter’s large
mass. Nevertheless, Newton treated the Sun-Earth-Moon system as a perturbation
of the two-body Earth-Moon system:17

The area which the moon describes by a radius drawn to the earth is
proportional to the time of description, excepting in so far as the moon’s
motion is disturbed by the action of the sun, and here we propose to
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investigate the inequality of the moment, or horary increment of that

area or motion so disturbed.

He specifically stated that the plane of the orbit oscillates and that the eccentricity
also varies. Even so, the theory of the Moon that he published in his Principia
was “very imperfect”, as he himself stated in the preface to this work. In order
to produce a better theory, Newton undertook a more general study of the three-
body problem but immediately ran into major difficulties. As a result he adopted
a simplified form of this problem either by choosing a 1/r form of the gravitational
law in order to simplify the computations, or else by assuming a very massive central
body whose motion was almost unaffected by the motion of the two other bodies,
or else taking one of the three bodies very far away from the other two. The general
problem was treated in a not very clear way in 22 corollaries in which he proposed
a semi qualitative evolution of the motion. Even though Newton’s mechanics had
already explained tidal phenomena and the flattening of the Earth at its poles, it
was unable to solve the problem of describing the motion of three bodies under their
mutual gravitational interaction.
Because it is very simple to state but also because it is at the heart of the theory
of the Moon, the three-body problem has been at the center of mathematical and
mechanical studies for close to two centuries.

17 I. Newton, The mathematical principles of natural philosophy (1687), Translated by A. Motte,

Book iii, Proposition xxvi, 1846.