Leonhard Euler and the Foundations of Celestial Mechanics

228 Pages·2022·6.216 MB·English

by  Dora Musielak

History of Physics
Dora Musielak
Leonhard Euler and
the Foundations
of Celestial
Mechanics
History of Physics
Series Editors
Arianna Borrelli, Institute of History and Philosophy of Science, Technology, and
Literature, Technical University of Berlin, Berlin, Germany
Olival Freire Junior, Instituto de Fisica, Federal University of Bahia, Campus de O,
Salvador, Bahia, Brazil
Bretislav Friedrich, Fritz Haber Institute of the Max Planck, Berlin, Berlin,
Germany
Dieter Hoffmann, Max Planck Institute for History of Science, Berlin, Germany
Mary Jo Nye, College of Liberal Arts, Oregon State University, Corvallis, OR,
USA
Horst Schmidt-Böcking, Institut für Kernphysik, Goethe-Universität, Frankfurt am
Main, Germany
Alessandro De Angelis , Physics and Astronomy Department, University of
Padua, Padova, Italy
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Dora Musielak
Leonhard Euler
and the Foundations
of Celestial Mechanics
Dora Musielak
Department of Physics
University of Texas
Arlington, TX, USA
ISSN 2730-7549 ISSN 2730-7557 (electronic)
History of Physics
ISBN 978-3-031-12321-4 ISBN 978-3-031-12322-1 (eBook)
https://doi.org/10.1007/978-3-031-12322-1
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A Tribute to the Blind Mathematician
who Saw Infinity
Portrait of Leonhard Euler by J. Darbès, 1778. It
depicts a wise scholar, the blind mathematician now
seeing beyond the stars. Source License: Public
Domain Mark https://wellcomecollection.org/works/
zgsvb5nh
Did you know?
1. Euler revolutionized mathematics and physics and became one of the founders
of analytical celestial mechanics.
At age sixteen, he earned a Master’s degree (A.L.M. = Artium Liberalium
Magister) after delivering a speech (in Latin), on the natural philosophy of
Newton compared with that of Descartes. He had begun to build a splendorous
mathematical universe, to help us understand Newtonian physics and to describe
the Solar System using the power of analysis.
2. To Euler, we owe the mathematical representation of the second law of motion
in differential form.
In 1734, he wrote the first major treatise of Newtonian mechanics in analytical
form. In 1747, Euler introduced the second law differential equation of motion
expressed in three-dimensional Cartesian coordinate components.
3. At nineteen, Euler made his first international voyage, from Basel (Switzerland)
to St. Petersburg (Russia), a linear trek of about 2650 km.
At the Imperial Academy of Sciences, Euler established a solid foundation in
mechanics, celestial mechanics, and mathematical physics.
4. Euler contributed to all areas of mathematics and physics.
He wrote seminal memoirs and books in astronomy, calculus, fluid dynamics,
hydraulics, hydrostatics, hydrodynamics, magnetism, mechanics, number
theory, music and sound, and in optics. He also dealt with applied fields that
involved the development of technologies such as artillery, cartography, civil
architecture, water pumps, and naval science.
5. Euler built his own apparatus to observe solar and lunar eclipses from his home.
I call him “blind astronomer” because by 1738 Euler had lost vision in one eye.
By 1770, he became completely blind. And yet Euler never stopped developing
mathematical theories to advance the field of celestial mechanics.
6. Euler was multilingual and a Universalist scholar.
A native speaker of Swiss German, Euler spoke Latin as a second tongue. He
wrote his memoirs in all scientific languages of his time: Latin, French, German,
vii
viii Didyouknow?
and he learned to speak and write in Russian. He translated a book from English
into German and also made contributions to the field of articulatory phonetics.
7. Euler’s mathematical productivity increased with age, despite his blindness.
Between the ages of 60 and 76, Euler authored the greatest number of memoirs,
over 60 percent of the total he published.
8. Before he died, Euler modeled the motion of a hot air balloon, without ever
seeing one.
His slate was found after his death, with the analysis modeling the motion of
aerostatic balloons. This he did after he heard about the manned flight in France
of the first air balloon designed by the Montgolfier brothers.
9. Orbital mechanics of artificial satellites and spacecraft is based on Euler’s
analysis of astromechanics.
Artificial satellite attitude and dynamics are founded on Euler’s rigid body
mechanics. Euler’s angles (which describe the orientation of a satellite’s orbit
in 3D), Euler’s equations of motion, and Euler’s theorem of rigid bodies are just
a few of the concepts we use today to analyze the time derivatives of spacecraft
moving state vectors.
10. Euler revolutionized the study of Newtonian mechanics by introducing for
the first-time infinitesimal calculus into the study of physics and celestial
mechanics.
The analytical mathematical methods that form the foundation of the engineer’s
and physicist’s curricula were developed by Euler. Engineers and physicists owe
so much to Euler!
Preface
Who does not know Euler, the incomparable mathematical genius? Whether we are
mathematicians, physicists, or engineers, we all have studied and used the theorems,
analytical methods, and mathematical tools he discovered; we even use the notation
he invented to write differential and integral equations with elegance and mathemat-
ical beauty. Yet, how many know Euler the theoretician astronomer, the physicist
who helped erect the analytical foundations of celestial mechanics?
I was motivated to write this book after realizing that Leonhard Euler was not fully
recognized for his pioneering role in expressing Newtonian celestial mechanics in
the language of analysis. When I studied the field, references to Newton’s Principia
were typically followed by Laplace’s Traité de mécanique celeste. In the study of
planetary or spacecraft orbits, only the work by Lagrange was cited when alluding
to the orbital equilibrium points in the Sun–Earth–Moon system. Why? To Euler,
we owe the whole mathematical apparatus that makes possible today to calculate
planetary and spacecraft orbits and much more.
At the turn of the eighteenth century, the structure, motion, and dynamic evolution
of our Solar System were explained in geometrical terms. With his Principia Math-
ematica, Isaac Newton had contributed a System of the World, the most advanced
book that provided a comprehensive organization of the heavens according to the
law of universal gravitation.
Astronomers observed and took ample measurements of the motions of heavenly
bodies, but none of them could describe accurately the dynamics of the Solar System,
lacking a solid theoretical basis for analysis. Then Leonhard Euler came along. With
his exquisite mathematical flair, Euler began to construct the foundation for the
rigorous study of the dynamical systems formed by the planets, moons, and comets,
based on both his profound knowledge of Newtonian mechanics and his vast ability
to connect mathematics to available astronomical data.
There were many puzzling phenomena still unexplained before Euler’s work.
When he was twenty years old, the Solar System contained only six planets, eleven
moons and perplexing, and sporadic comets. The motion of the Moon and the
planets exhibited inequalities (variations in their orbits caused by perturbations) that
ix
x Preface
astronomers could not explain. There was an urgent need for analytical methods to
quantify perturbations due to mutual gravitational interactions.
Leonhard Euler contributed to advance a wide spectrum of topics in celestial
mechanics. At the St. Petersburg Observatory, Euler observed sunspots and tracked
the movements of the Moon. As his research advanced, Euler combined astronomical
observations with his own mathematical genius and determined the orbits of planets,
Moon, and comets. Euler conceived the methods of planetary perturbations, solving
many of the Newtonian mechanics problems of the eighteenth century that are still
relevant today.
The scope of Euler’s research spans the entire range of astronomical problems
that were posed by Newton. This includes analysis of orbit perturbations, planetary
and comet motion, lunar theory, eclipses, tides, and work on optics, and refinement
of lenses for telescopes.
It was Euler who first formulated equations of motion of the three-body problem
for the Earth–Sun–Moon system in differential form. In his pioneering study of the
three-body problem, which Newton did not solve, Euler discovered the collinear equi-
librium points in a restricted three-body system, which became of crucial importance
for the advancement of space exploration in the twentieth century. Euler’s work in
mathematical astronomy was crowned with six of the twelve prizes he won from the
French Academy of Sciences.
Euler was the first to use the trigonometric series in celestial mechanics and
discovered the method of undetermined coefficients to integrate the series, just as
we do it today. He was the first to use trigonometric functions in perturbation theory.
Other mathematicians, including Clairaut, d’Alembert, Lagrange, and Laplace, used
Euler’s series expansions in their analysis to deal with planetary perturbations.
Euler built the mathematical foundation of analysis and contributed to all aspects
of pure and applied mathematics. He worked in exponential functions and intro-
duced the familiar symbol e for the transcendental number 2.7182. His monumental
Opera Omnia is rather extensive and full of scientific jewels. There is much more in
analytical methods that Euler conceived, leaving a significant legacy for advancing
the study of celestial mechanics.
It is not possible to do justice to the overwhelming range and depth of Euler’s work
on celestial mechanics in a single book. However, inspired by some of what Euler
did in total darkness (yes, he was blind), I felt compelled to promote at least what
Euler did that contributed to making possible humanity’s space exploration program,
activities we pursue today with human missions, space telescopes, and space probes
to expand our reach to the heavens.
This book is organized as follows: Chapter 1 is devoted to tracing Euler’s early
life and career. Not intended as a biography, I will simply highlight the status of
mathematics and science at the time in which he was born. Chapters 2 and 3 chronicle
Euler’s journey from Basel, Switzerland to Saint Petersburg in Russia, and his first
activities at the Russian Academy where he conducted research closely related to
astronomy. Chapter 4 is devoted exclusively to the first theory of sea tides that Euler
developed based on Newton’s law of gravity in response to a prize competition. In
Chapter 5, I will highlight the various areas of research Euler pursued after joining