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LAGRANGIAN MECHANICS LECTURE NOTES
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Lecture notes of 133 pages for the course Computational And Applied Mathematics at wits (SUB TOPIC OF CAM)
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L AGRANGIAN M ECHANICSAPPM2007
Course Notes by
D. Fanucchi, W.A. Carlson and M.J. Woolway
2019
,
,Preface
This is the set of course notes for APPM2007. The notes are intended to complement the
lectures and other course material, and are by no means intended to be complete. Students
should consult the various references that have been given to find additional material, and
different views on the subject matter.
This material is under development. Please would you report any errors or problems (no
matter how big or small). Any suggestions would be gratefully appreciated.
School of Computer Science and Applied Mathematics,
University of the Witwatersrand,
Johannesburg,
South Africa
This work is licensed under the Creative Commons
Attribution-NonCommercial-NoDerivs License. To view a copy of this license, visit,
• http://creativecommons.org/licenses/by-nc-nd/2.5/
Send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California,
94105, USA. In summary, this means you are free to make copies provided it is not for
commercial purposes and you do not change the material.
i
, Foreword
We live today in an era of discovery, where new phenomena are constantly being observed
and explained, where thousands of scientists are competing in an undying quest to discover
the secret language of the universe. We live in a world of intellectual abundance, a world
where precious gems of knowledge, mined for so conscientiously by our predecessors and
contemporaries are just a mouse-click away. It is certainly exciting to witness this lightning-
quick development of human understanding, which expands into new domains with more
pace and agility than the most riveting game of sports.
What, then, might lead us to the next breakthrough? What might decode the beautiful
language of that grand and silent masterpiece - the universe? Perhaps it is prudent to heed
the words of Newton that we can only progress by "Standing on the Shoulders of Giants".
With this inspiration, let us embark on a journey of the mind to the very origins of our
scientific understanding. Our journey begins in the year 2010 and we travel slowly
backwards, noticing as we go the thousands of Journal Articles and PhD’s published in the
time we are winding back. Soon we have passed to a time before social networking was a
concept, and before cell phones had colour screens. Our journey begins to speed up now,
and we see the information revolution and the internet disappearing. Let us go further still,
before humans had travelled into space, before television, before the Wright Brothers
developed the first Aeroplane, before radio was discovered, before highways and
skyscrapers and cars and trucks. Let us go back to before Charles Babbage first conceived
his Analytical Engine, and continue still, to before electricity was understood and before
railways were ever built. Now we are travelling at a breathtaking speed beyond the first
telescope, crafted by Galileo Galilei, and beyond the first use of a magnetic compass. We
are reaching the furthest stretches of known history - around us is a desolate plane. Human
beings are living in caves and hiding out in the vast unknown. Let us pause here, and begin
our walk back to the present.
Humans, from the very outset, have been driven by the urge to understand the world.
Even in the distant past we observe the prehistoric man carving out a tool from rock and
wielding it to best his prey. Further forward we see the discovery of fire, the taming of
wolves and the logging of the stars above. We now jump to the time of the Ancient Greeks.
We see sundials and water-watches for telling the time of day, pullies and levers for moving
ii
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