History of Mechanics

The history of mechanics, including statics, hydrodynamics, astronomy and terrestrial, commonplace dynamics
appears convoluted and perhaps incomplete.

From our texts we get the idea that Newton, pretty much in the same style as Moses, declared
three revealed laws, that went on to herald an age of prosperity and enlightenment. Mechanics certainly
did not begin with Newton (or end with him). Not even the Aristotle-> Dark Ages-> Copernicus->Kepler->Galileo
->Newton picture is accurate. The Dark Ages were only in Western Europe. It was pretty bright elsewhere, especially
in the Arab/Persian/Moorish world. So there are now histories that involve Arabs, "Hindus", Chinese apart from the usual Greek
contributions. Surely there are national and other prejudices that colour any version of history, but even factoring this, it appears that
we take a risk of erasing large parts of history if we ignore them.

Not being up on this at all, I did the usual googling and here is a bit of what I gathered.

I thought this small article and discussion that follows it is very interesting:
John Baez's very brief account of the history

The Wiki entry on history of classical mechanics mentions important Islamic contributions around 10 AD,
from Al Biruni, Al-Khazini, Ibn Sina (Avicenna) and several others.
Wikipedia on History of Classical Mechanics

These links and the many others that you can follow from here will give a picture of the exciting way the
view of the world of motion developed, the view that we today take for granted. For instance the first law of
Newton is a culmination of many centuries of observation, debate and very very importantly: abstraction. For there
can never be systems that approach the isolation that is required, yet friction is considered a "second level" phenomenon,
a collective or effective phenomenon. This abstraction did wonders for reductionist science as it allowed to
concentrate on the simplest and the most essential, it postulated a picture of a complex world made of simple bocks.
While mechanics itself has been challenged "externally" through subsequent developments of quatum mechanics
and relativity which challenged many basic givens, it has undergone several changes "internally" as well. It is
now recognized that the determinism that is built into classical physics is not of practical use in most situations as
it can give rise to "deterministic chaos". The pendulum played a very important role in the development of mechanics
as Galileo found its time period to be independent of its amplitude. It became a symbol of a very orderly universe,
indeed the "clockwork universe". But attach another pendulum to the end of one, we get the double pendulum and
the motion of this is far from clockwork. See here for an animation of the double pendulum

This brings us to modern developments in classical mechanics. How well do we understand the erratic looking motion of
the double pendulum? Most people probably do not appreciate that there are open problems in classical mechanics, simple to state,
important open problems. These are not in the nature of formulating new theories of mechanics, but understanding what the theories
imply. Some of the histories of mechanics do not mention the seminal contributions of Poincare, Birkhoff, Kolmogorov,
Arnold, or Moser made during the last one hundred years.

Let me summarize my meager understanding of this history now.

Era of Epicycles

Going back, maybe we can divide developments into certain historical segments.
First was the era that culminated with Ptolemy around 160 A.D. whose book and
table Almagest was in use to predict accurately the positions of the planets, the sun, moon and stars for nearly 1400 years.
Indeed a very remarkable achievement for a "wrong" model of the solar system. The model was a geocentric one with planets
and the sun fixed on rotating spheres or circles called epicycles. The number of such epicycles varied and about 40 of them
were needed to explain the retrograde (back and forth, west to east and back) motion of some planets like Mars. In a very distant
final sphere were the stars. For an animation of epicycles take a look at the website
This site also has nice discussions of the early developers and further animations. Strongly recommended!

I believe many of us who have not followed the motion of any planet, think of epicycles as silly. We with our superior
understanding of the solar system and much much beyond. On the other hand Ptolemy's epicycles (which predates him)
is an extremely sound idea and is in essence the first application of the Fourier series. The length of the cycles were
the coefficients in the series and the angular velocities the principal frequencies. Thus the fact that the orbits were not
perfect circles implied that more than one frequency was needed.

During this era, apart from the position of the "heavenly" objects there were very important developments in statics that are ascribed to
Archimedes and the whole body of Aristotle's work that is based on a strong sense of logic and commonsense. For instance his idea that velocity is proportional to the force and inversely to some resistance dominated for several centuries, when the focus slowly shifted from velocity to acceleration.

The Era of Islam and the East

The second era seems dominated by the Pre-Islamic, and later, Islamic scholars who translated the greek texts into Arabic and
Latin and were spread in what is perhaps today middle east, Iran, and northern Africa. At this same time
rather remarkable developments happened in India, with Aryabhatta and his followers around the 5th century AD.
This included development of very accurate calculations of the eclipses, development of the sine and cosine functions (and their tables)
and of course a lot of algebra, such as quadratic equations etc.. However I have seen conflicting statements about whether he was
a "geocentrist" or a "Heliocentrist". From my reading I would opt for the former. Here's an article by Prof. Ansari on Aryabhatta's life and
works aryabhatta.pdf. You can download the aryabhattiya text (and many more like those of Ptolemy, Euclid and others!) from http://www.wilbourhall.org/index.html
About the Islamic contributions, it is said that polymaths like Al Biruni and Avicenna deviated radically from Aristotle in their
reliance on empirical evidence (read experiments) rather than logic. Read more of their contributions from the Wiki referred to above
and the links therein (therefrom?).

The era of Kepler and Newton

The Third era is undoubtedly the most important and has the monk Copernicus who reverted to the Heliocentric picture
thus kicking off festivities during the 15th century. Copernicus was also a polymath, an eminent physician, an eminent and much sought after
economist as well as an amateur astronomer. He is said to have drawn from works by Islamic scholars, especially Averroes
who had a lasting influence on Europe, through his translations of Aristotle into Latin around the 12th century. While Copernicus shifted the
sun to the center, he retained circular orbits and thus still did worse than Ptolemy. He had to retain epicycles even in the Heliocentric
picture to do well. Next comes the genius and personality of Johannes Kepler. At that time Astrology and Astronomy
were more closely allied than Physics and Astronomy, and Kepler (who incidentally was also an astrologer) made the first marriage between
these. The five Platonic Solids inscribed and circumscribed by spheres were the first model that an inspired Kepler put out. Kepler thought he had heard the music of the heavens and had fathomed a bit at least of the mind of God. But he realized that this was not so, and later struck up a professional relation with Tycho Brahe, whose extensive astronomical data he analyzed.

Within Kepler's religious view of the cosmos, the Sun (a symbol of God the Father) was the source of motive force in the solar system. As a physical basis, Kepler drew by analogy on William Gilbert's theory of the magnetic soul of the Earth from De Magnete (1600) and on his own work on optics. Kepler supposed that the motive power (or motive species) radiated by the Sun weakens with distance, causing faster or slower motion as planets move closer or farther from it. Perhaps this assumption entailed a mathematical relationship that would restore astronomical order. Based on measurements of the aphelion and perihelion of the Earth and Mars, he created a formula in which a planet's rate of motion is inversely proportional to its distance from the Sun. Verifying this relationship throughout the orbital cycle, however, required very extensive calculation; to simplify this task, by late 1602 Kepler reformulated the proportion in terms of geometry: planets sweep out equal areas in equal times — the second law of planetary motion.

He then set about calculating the entire orbit of Mars, using the geometrical rate law and assuming an egg-shaped ovoid orbit. After approximately 40 failed attempts, in early 1605 he at last hit upon the idea of an ellipse, which he had previously assumed to be too simple a solution for earlier astronomers to have overlooked. Finding that an elliptical orbit fit the Mars data, he immediately concluded that all planets move in ellipses, with the sun at one focus — the first law of planetary motion. Because he employed no calculating assistants, however, he did not extend the mathematical analysis beyond Mars. By the end of the year, he completed the manuscript for Astronomia nova, though it would not be published until 1609 due to legal disputes over the use of Tycho's observations, the property of his heirs. From the Wiki on Kepler

Thus Kepler after a remarkable journey formulated his three laws which were, much to his disappointment, ignored by his
famous contemporaries Galileo Galilei and Rene Decartes. While Galileo
made several well known contributions to the foundations of mechanics, he refused to accept the elliptical orbits of Kepler,
voting for the circular orbits of Copernicus, due to the notion that the circle was somehow "perfect". It took Sir Isaac Newton
of course to show that an inverse square gravitational force combined with the second law of motion implied Kepler's three laws of
motion. Thus the connection in the Ptolemy-Kepler-Newton triad is very strong, although it must also be said that the
inverse square force was very much in the air at the time of Newton. By co-founding calculus (along with Leibniz) he laid the
foundation of a very remarkably successful endeavor that is essentially all of modern sciences. He also joined "heavens" and the
earth by realizing that the same laws that apply on and to the lowly, imperfect earth, are also operative in the heavens.
Till that time Astrology and Astronomy were inseparable and thought to be independent of physics.

The era of Action

The fourth era.
Newtonian Mechanics was reformulated subsequently, facilitating solutions of more complex problems, developing
perturbation theories, uncovering underlying extremal principles, and revealing a unique mathematical structure (called symplectic geometry)
D'Alembert's principal of virtual work, which effectively states that the forces of constraint do no work, played a pivotal role in Lagrange's (Joseph Louis Lagrange) formulation of mechanics called Lagrangian Mechanics. These developments happened about 100-150 years after Newton in the late 18th and early 19th century. Along with Euler, Lagrange founded the theory of calculus of variations and it was natural that he perhaps looked for a formulation of classical mechanics that was based on an extremal principle. He was building on earlier work done by Maupertuis who argued for a principle of "Least Action". The action of Maupertuis was a product of momentum and the distance travelled. That this was 'minimised' was put forward by Maupertuis as an economy principle which implied the efficient design of God.
It is interesting that most of the early developers of Mechanics, including Newton (who wrote more on Religion than on science), saw God as
behind the designs that were discovered by them. That there was a mathematical organization was taken by them to imply an higher intelligence.

Lagrange's reformulation of mechanics did away with the need to explicitly consider forces of constraints. Those pesky "tensions" and "normal forces" were found to be added baggage. Lagrange called his the "Analytical Mechanics" and was famously proud that his work had no diagrams
at all. M.C. Gutzwiller in his book on chaos says that even Lagrange did not write down what we today called
the Lagrangian. This was done by William Rowan Hamilton in the mid 19th century. Having made
seminal contributions to Geometrical optics, Hamilton reformulated classical mechanics in terms of a principle of extremal action, introduced
the phase space as the natural space of mechanics and wrote his famous equations. This was to form the mathematical stage that was
most natural to develop both statistical mechanics which lies at the foundation of thermodynamics and quantum mechanics.
The "action" plays a central role in modern physics, such as in quantum field theory. It is the fundamental quantity from which all else follows.

The era of Chaos

The fifth era begins with Henri Poincare, the last of the mathematical universalists and one of the last polymaths.
In relation to mechanics, his most important contribution was his study of the three-body-problem. The problem of the stability of the
solar system was (and still is) an open problem. Poincare's work on this led to the discovery of the homoclinic tangle, which is at the
heart of what today is called "chaos" and what we saw above in the erratic motion of the double pendulum. Hamiltonian chaos is part
of a general phenomenon of chaos that has come to be widely appreciated only after the advent of computers in the 1960s and
1970s. Popularized in the public imagination as the "butterfly effect" it is about how sensitive dynamical
evolution is to changes in the initial conditions. Behind a plethora of mesmerizing pictures of strange attractors and Mandelbrot set is
a very sober mathematical world of dynamical systems or nonlinear dynamics with important ramifications for our understanding of the
universe. The fifth era is the era of chaos.

Getting back to the history, Poincare also developed important techniques to study nonlinear perturbations. Following Poincare, is the American mathematician and physicist George Birkhoff who laid foundations to general dynamical systems theory and elucidated what is known as the Poincare-Birkhoff theorem which is about the fate of rational tori in under perturbations. He also made
very important contributions to ergodic theory, which connects classical and statistical mechanics. The last of this very brief and hopelessly
inadequate "history" is the KAM theorem of A. Kolmogorov, V.I. Arnold and J. Moser, formulated in the mid 1950s and early 1960s. They showed that quasiperidic (non-resonant) motions persist under sufficiently small and smooth
perturbations. Being one of the most difficult theorems to prove, it does not form part of any syllabus! There are many related open
problems. For instance the Poincare-Birkhoff theorem, mentioned above, is restricted to two-degrees of freedom, the generalization
to higher dimensions being unknown. The history of classical mechanics is not yet over.

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