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An Introduction to Celestial Mechanics
Classic text still unsurpassed in presentation of fundamental principles. Covers rectilinear motion, central forces, problems of two and three bodies, much more. Includes over 200 problems, some with answers. Reprint of revised 1914 edition.
464 pages, Paperback
First published January 1, 1914
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October 26, 2023
It always pays to go back and read a classic textbook from a century ago or more. One learns to see through another’s eyes. When we think about celestial mechanics these days, uppermost in our minds will be the Kolmogorov-Arnold-Moser theorem and the modern theory of chaotic dynamical systems, as it has emerged in the past half-century or so. But the present work by Forest Ray Moulton, An Introduction to Celestial Mechanics, originally written in 1902 and conveniently available as a Dover reprint since 1984, knows nothing of such things.
Moulton was a known expert in theoretical astronomy at the University of Chicago, dependable if not perhaps as incandescently brilliant as, say, his contemporary Henri Poincaré. Rather than original and pioneering novel ideas and perspectives, as found in Poincaré’s Les Méthodes Nouvelle de la Mécanique Céleste from 1893, what one gets here is a balanced treatment of all the traditional aspects of the field as it would have appeared around the turn of the twentieth century, undistracted in its focus by modern dynamical systems theory and its typical concerns. Therefore, excellent material for the beginner!
For here one will find a straightforward introduction to Newtonian classical mechanics, covering the equation of motion subject to various force laws (central, friction proportional to velocity, inverse square of other dependence on distance) or how to find the potential and attraction at the surface of an oblate spheroid or a circular disc and so forth, along with some dated topics such as Helmholtz’ theory of gravitational contraction as the source of the sun’s luminosity.
But for most anyone who would take the trouble to look up an old text such as this, its real interest must center upon Moulton’s treatment of what we would regard as the problems proper to celestial mechanics per se, to which he turns from chapter five onwards. As one would expect, a clean exposition of the two-body problem and methods of solution to Kepler’s equation (which one has to solve to determine where on its orbit the planet will be located at any given time; since it is transcendental there is no closed-form analytical expression and, in an era before digital computers, a lot of effort was expended on approximate numerical approaches such as series expansions in the ellipticity – which, of course, hold good for most of the planets, whose orbits are nearly circular). Next, a practical discussion of coordinate systems, a topic usually passed over these days by everyone except for specialists who have to carry out actual computations: the heliocentric position in the ecliptic system (ascending and descending node, inclination, longitude of the perihelion, argument of the latitude) and transfer of the origin to the earth in geocentric equatorial coordinates (what the observer would actually use). No fancy mathematics here, but it’s nice to see everything written out clearly in case one would ever need to refer to it.
Chapter six covers the problem of determination of orbital elements from the apparent positions. This reviewer finds the subject quite interesting, since, although it is evident in principle how one would have to proceed, it is not necessarily so easy to perform all the algebraic steps in an efficient manner. By Moulton’s day, needless to say, all the thought devoted to the problem over previous centuries had issued in a fairly streamlined procedure which he explains with his usual clarity (Laplacian and Gaussian methods).
At last in chapters seven and eight, we arrive at the three-body problem: how to simplify it by taking advantage of the integrals of motion (including Jacobi’s integral in rotating coordinates), general characteristic of the phase-space portrait, Tisserand’s criterion for identifying comets and a straightforward stability analysis of the Lagrange points.
The last two chapters have to do with perturbative methods – Hill’s lunar theory in chapter nine and the analytical approach to the general solar system in chapter ten. If one wants just a little taste of how all this goes, Moulton’s treatment is exemplary and comparatively simple, in view of the inherent complexity of the problem. For instance, an easy-to-follow explanation of the evection of the moon’s orbit (a periodic variation in eccentricity which accounts for the largest deviation in magnitude and was known to Hipparchus and Ptolemy). In chapter ten, the theorem on the stability of the solar system (absence of secular terms up to first order) – already complicated enough, but it is truly remarkable that Laplace in 1773 showed this to second order, at the tender age of twenty-four!
There are other textbooks from which to acquire facility with more advanced techniques (such as the use of canonical transformations in place of Lagrange brackets). If one is serious about pursuing perturbations to second order (in the eccentricities, in the masses, in the inclinations etc.), modern approaches are invaluable in keeping the requisite number of terms to a minimum – that astronomers in Moulton’s day got as far as they did without enjoying any such advantage and without digital computers, to boot, is most impressive! But there is something to be said for first learning the ropes by means of a simpler approach to the subject – though even Lagrange’s, as covered here, goes well beyond what would immediately suggest itself if one were to attempt to tackle the problem de novo.
Therefore, this classic work by Moulton deserves a recommendation as an entryway into the subject. Those who aspire to become proficient will not be long detained by perusing it, while the amateur may at least gain from it a healthy appreciation in more than superficial detail of the monumental accomplishments that have been made in celestial mechanics since Newton – what Moulton himself, in his closing reflections, praises as ‘entitled to be regarded as the most perfect science and one of the most splendid achievements of the human mind’.
Moulton was a known expert in theoretical astronomy at the University of Chicago, dependable if not perhaps as incandescently brilliant as, say, his contemporary Henri Poincaré. Rather than original and pioneering novel ideas and perspectives, as found in Poincaré’s Les Méthodes Nouvelle de la Mécanique Céleste from 1893, what one gets here is a balanced treatment of all the traditional aspects of the field as it would have appeared around the turn of the twentieth century, undistracted in its focus by modern dynamical systems theory and its typical concerns. Therefore, excellent material for the beginner!
For here one will find a straightforward introduction to Newtonian classical mechanics, covering the equation of motion subject to various force laws (central, friction proportional to velocity, inverse square of other dependence on distance) or how to find the potential and attraction at the surface of an oblate spheroid or a circular disc and so forth, along with some dated topics such as Helmholtz’ theory of gravitational contraction as the source of the sun’s luminosity.
But for most anyone who would take the trouble to look up an old text such as this, its real interest must center upon Moulton’s treatment of what we would regard as the problems proper to celestial mechanics per se, to which he turns from chapter five onwards. As one would expect, a clean exposition of the two-body problem and methods of solution to Kepler’s equation (which one has to solve to determine where on its orbit the planet will be located at any given time; since it is transcendental there is no closed-form analytical expression and, in an era before digital computers, a lot of effort was expended on approximate numerical approaches such as series expansions in the ellipticity – which, of course, hold good for most of the planets, whose orbits are nearly circular). Next, a practical discussion of coordinate systems, a topic usually passed over these days by everyone except for specialists who have to carry out actual computations: the heliocentric position in the ecliptic system (ascending and descending node, inclination, longitude of the perihelion, argument of the latitude) and transfer of the origin to the earth in geocentric equatorial coordinates (what the observer would actually use). No fancy mathematics here, but it’s nice to see everything written out clearly in case one would ever need to refer to it.
Chapter six covers the problem of determination of orbital elements from the apparent positions. This reviewer finds the subject quite interesting, since, although it is evident in principle how one would have to proceed, it is not necessarily so easy to perform all the algebraic steps in an efficient manner. By Moulton’s day, needless to say, all the thought devoted to the problem over previous centuries had issued in a fairly streamlined procedure which he explains with his usual clarity (Laplacian and Gaussian methods).
At last in chapters seven and eight, we arrive at the three-body problem: how to simplify it by taking advantage of the integrals of motion (including Jacobi’s integral in rotating coordinates), general characteristic of the phase-space portrait, Tisserand’s criterion for identifying comets and a straightforward stability analysis of the Lagrange points.
The last two chapters have to do with perturbative methods – Hill’s lunar theory in chapter nine and the analytical approach to the general solar system in chapter ten. If one wants just a little taste of how all this goes, Moulton’s treatment is exemplary and comparatively simple, in view of the inherent complexity of the problem. For instance, an easy-to-follow explanation of the evection of the moon’s orbit (a periodic variation in eccentricity which accounts for the largest deviation in magnitude and was known to Hipparchus and Ptolemy). In chapter ten, the theorem on the stability of the solar system (absence of secular terms up to first order) – already complicated enough, but it is truly remarkable that Laplace in 1773 showed this to second order, at the tender age of twenty-four!
There are other textbooks from which to acquire facility with more advanced techniques (such as the use of canonical transformations in place of Lagrange brackets). If one is serious about pursuing perturbations to second order (in the eccentricities, in the masses, in the inclinations etc.), modern approaches are invaluable in keeping the requisite number of terms to a minimum – that astronomers in Moulton’s day got as far as they did without enjoying any such advantage and without digital computers, to boot, is most impressive! But there is something to be said for first learning the ropes by means of a simpler approach to the subject – though even Lagrange’s, as covered here, goes well beyond what would immediately suggest itself if one were to attempt to tackle the problem de novo.
Therefore, this classic work by Moulton deserves a recommendation as an entryway into the subject. Those who aspire to become proficient will not be long detained by perusing it, while the amateur may at least gain from it a healthy appreciation in more than superficial detail of the monumental accomplishments that have been made in celestial mechanics since Newton – what Moulton himself, in his closing reflections, praises as ‘entitled to be regarded as the most perfect science and one of the most splendid achievements of the human mind’.
June 27, 2024
Much of the development of calculus and differential equations was due to the desire to explain the motion of the planets. This book gives some excellent historical examples of how this all progressed. However, this book is not particularly accessible to those who don’t want to delve into the math.
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