Celestial Mechanics
J. Massimino
History of Mathematics
Rutgers, Spring 2000
Throughout the history of mathematics, branches of scientific study have regularly used mathematical methods to explain natural phenomena. This is very true in the field of astronomy, and particularly in the case of celestial mechanics. Celestial mechanics is defined as the branch of astronomy dealing with the mathematical theory of the motions of celestial bodies. The curiosities of celestial mechanics date back in to ancient times. The ancient Greeks, for instance, gave divine status to the cosmos, and therefore felt the need for further exploration. At this time, astrology established itself and became a part of everyday life. The principle that stars could influence man's life "...received some kind of justification from the notion of cosmos, a cosmos which is so well arranged that no part is independent of the other parts and of the whole. Was this not proved by the tides, caused by Moon and Sun, by the menstruation of women, by the farmers' moonlore... (Sarton, 165)." During the third century BC, Aristarchos of Samos combined Euclidean geometry with the assumption that the Sun is the center of the "universe" rather than the Earth^{1}. In his model the planets circle the Sun, the Moon orbits the Earth, and the stars are fixed, while their apparent "rotation" is an illusion caused by the Earth's rotation. This heliocentric model was later reaffirmed by Nicholas Copernicus (c. 1520).
In the late 17th century, a new age of thinking, now called the Enlightenment, began. During this Age of Reason, intellectuals looked for a single principle that would join together the concepts of nature, God, and reason^{2}. Sir Isaac Newton (1642-1727) of Great Britain made a major contribution to the scientific world during this time with the publication of his greatest work, Philosophiae Naturalis Principia Mathematica, later referred to simply as Principia. In this great work Newton discussed many influential ideas, including his famous Laws of Motion, which dramatically affected the understanding of celestial mechanics. These laws are:
The Life of Joseph-Louis Lagrange^{4}
Joseph-Louis Lagrange was born on January 25, 1736 in Turin to an influential family. At the age of fourteen, Lagrange was sent to the University of Turin to study law, after his father went bankrupt after poor financial speculation. Although he was to study law, his interests and abilities quickly showed to favor mathematics, especially math analysis. Justly so, Lagrange was appointed substitute professor at the Royal Artillery School in Turin in 1755, and only two years later established the Royal T urin Academy of Sciences with his colleagues, chemist Count Saluzzo di Monesiglio (1734-1810) and anatomist Giovanni Cigna (1734-1790). However, in 1766, Lagrange grew unhappy with the limited research resources available in Turin, and so, moved to Berl in where he was installed as Director of Mathematics in the Berlin Academy. His years in Berlin proved to be the most fruitful period of his life. "During this time, he wrote on all branches of mathematics and intensively studied mechanics. The majorit y of his memoirs during this period deal with celestial mechanics (Lagrange, xix)." The writings from these memoirs were submitted to the Académie des Sciences de Paris, whose major scientific questions during the Enlightenment focused around:
With political changes occurring in Berlin in 1786, Lagrange again became unhappy and dissatisfied and so accepted the position of pensionaire vétéran of the Académie des Sciences, causing him to move to Paris. In less than a year later, the first e dition of Lagrange's most famous work, Mécanique Analytique, was published. This book dealt with his findings about principles of statics and dynamics and eventually laid the groundwork for further studies in mechanics. Lagrange continued his studies in Paris, despite the fact that the French Revolution (c. 1789) was all around him, even through to his death in 1813. Just prior to his death, Lagrange summarized his life with these words:
In the second edition of Mécanique Analytique, Lagrange's work focused more on celestial mechanics than the previous edition had. In this edition, Lagrange produced equations^{6} aiding the understanding of celestial mechanics, including those dealing with:
In the world system, according to Lagrange, the force of attraction is inversely proportional to the square of the distance.
R= g/r^{2} => R dr=-g/r
Using the equation between F and r, the substitution becomes
Planetary Periods
In order to compute the period of a planet, Lagrange began with the equation
Recalling that r(1+ecosq)=b
and r=a(1-ecosq),
Lagrange substituted
in X=a(cosq - e),
so that X= (b-r)/e = (a(1-e^{2})-r)/e. Therefore,
X=a(cosq - e) and
Up to this point, Lagrange had shown how to express the elements of
elliptical orbits using the functions of x,y,z and of their
differentials dx/dt, dy/dt, and dz/dt. However, it was observed that
planets were subject to impulses which affected the veloci ties.
This, Lagrange concluded, could be accounted for by the following
adjustments:
dx/dt --> dx/dt + x^{.}
dy/dt --> dy/dt + y^{.}
dz/dt --> dz/dt + z^{.}
This gave the new elements of the planetary orbit
after the impulse.
Then using the radius vector
r, y, and
r
Lagrange
rewrote the elements of the orbits as:
Although the previous equations worked for arbitrary, momentary impulses, Lagrange realized that it was necessary to compensate for impulses that are infinitesimal and continuous, or perturbation forces.
His work on perturbation forces begins:
Allowing the following definitions, dx/dt = x', dy/dt = y', dz/dt = z', Lagrange concluded that each of the elements of the orbit could be expressed using x, y, z, x', y', z'. If a is one of these elements, then it "... will have its variation da by augmenting x', y', z' of the infinitesimal quantities Xdt, Ydt, Zdt. Thus one will have da = (da/dx' X + da/dy' Y + da/dz' Z) dt and similar equations will be obtained for the other elements ... (Lagrange, 367)."
This work on the perturbation theory became a significant part of the future of celestial mechanics.
With the development of Lagrange's equations for perturbations, significant developments have occurred. It was through the use of perturbation theory that Neptune was discovered. When astronomers of the mid-nineteenth century observed a perturbed orbit of Uranus, "astronomers Adams and Leverrier independently came to the conclusion that the perturbation must be due to a planet as yet unknown to astronomers (Kramer, 221)." This planet was soon discovered and named Neptune. The discovery of Pluto in th e twentieth century was similar.
Celestial mechanics has developed greatly throughout many centuries of observation. From a time where the cosmos were used to explain daily activity to the time of the Enlightenment when knowledge of how the universe worked was desired, it is an ever re levant field. The work of Joseph-Louis Lagrange, during the Enlightenment, proved to be a prominent and fruitful assets to our current understanding of the world.
Table of Variables
Notes