Lectures on Celestial Mechanics

by Carl L. Siegel, Jürgen K. Moser, C.I. Kalme (Translator)

The present book represents to a large extent the translation of the German "Vorlesungen über Himmelsmechanik" by C. L. Siegel. The demand for a new edition and for an English translation gave rise to the present volume which, however, goes beyond a mere translation. To take account of recent work in this field a number of sections have been added, especially in the third chapter which deals with the stability theory. Still, it has not been attempted to give a complete presentation of the subject, and the basic prganization of Siegel's original book has not been altered. The emphasis lies in the development of results and analytic methods which are based on the ideas of H. Poincare, G. D. Birkhoff, A. Liapunov and, as far as Chapter I is concerned, on the work of K. F. Sundman and C. L. Siegel. In recent years the measure-theoretical aspects of mechanics have been revitalized and have led to new results which will not be discussed here. In this connection we refer, in particular, to the interesting book by V. I. Arnold and A. Avez on "Problemes Ergodiques de la Mecanique Classique", which stresses the interaction of ergodic theory and mechanics. We list the points in which the present book differs from the German text. In the first chapter two sections on the tri pie collision in the three­ body problem have been added by C. L. Siegel.

Genres: Physics

302 pages, Paperback

First published November 18, 1971

Reviews

[William Bies · Rating 5 out of 5]

For the hard-core afficionado of celestial mechanics, this is the book everyone else refers to for full details of the proofs! C.L. Siegel was a capable mid-twentieth-century German astronomer who obtained impressive results in a wide range of fields in modern mathematics and, in particular, became a founder of the modern theory of dynamical systems; Jürgen Moser was a doctoral candidate under Siegel in the 1950’s who went on to clinch the proof of the KAM theorem in 1962. In 1971, he decided to update Siegel’s classic Vorlesungen über Himmelsmechanik (volume 85 in Springer-Verlag’s prestigious Grundlehren der mathematischen Wissenschaften, originally published in 1956) and to issue it in English translation. In recognition of the merit of the revised edition, Springer-Verlag has reprinted Siegel and Moser’s Lectures on Celestial Mechanics in its paperback series, Classics in Mathematics, in 1995.

After formally setting up the N-body problem, the first results of significance to be covered are those originally due to Sundman concerning collisions: a direct collision among all N bodies (as opposed to a very close approach) can happen only if the total angular moment equals zero. The method of proof consists in deriving Lagrange’s formula for the second derivative of the moment of inertia and cleverly exploiting its positivity. Further statements follow if one limits oneself to the case of three bodies, namely, if all three are to collide, their motion must take place in a fixed plane.

The next topic is regularization. There, one wants to change variable so that the singularity is removed and one can follow the motion through the moment of collision. The regularized representation is then applied to the three-body problem, reviewing further estimates of Sundman’s and showing existence of a convergent power series for all time in the uniformizing parameter. Analyticity in the strip follows from estimates plus the existence theorem for ordinary differential equations, map conformally to a disk to get the new time parameter. Show existence of power series expansion for all real time. In particular, zeros cannot accumulate at any finite time but they can at infinity, though there is a lower bound for time between successive collisions (from the Cauchy integral formula).

The last section in chapter one is devoted to showing existence of triple-collision orbits in special cases (equilateral and collinear). A convergent series expansion for triple-collision orbits (not practical for computation of course). Taking advantage of the canonical transformation to appropriate moving Cartesian frame to obtain a lot of integrals of the motion, reducing to the differential equation (13.28) subjected to linear stability analysis, having essential singularity if exponents are irrational (true apart from exceptional values of μ and ν depending on m1, m2, m3).

Chapter two takes up a classical topic that has seen renewed life in recent decades (not covered in this volume). Show that here do not exist any stationary solutions of the N-body problem; for a uniformly rotating frame, derive the Lagrange points L1-5. Linear stability theory, applied to Hamiltonian systems (to bring S into normal form). Analytically continue in the parameter to find a family of periodic solutions in non-linear case, employ the method power series with undetermined coefficients. It may be surprising that such a direct method leads to results: one can show existence of a period solution if the series converges (proof by method of majorants) but if there is an instability there could still be a periodic solution though proof of it becomes a global question inaccessible to power series methods (example on p. 110).

Apply these techniques to the planar three body problem near Lagrange points – nice because one can handle explicit expressions for the eigenvalues. The hard part is to get rid of multiple +/– i eigenvalues by reducing the Hamiltonian systems (Jacobi’s elimination of the node, NB center of mass integrals are time-dependent in rotating coordinates and cannot be used): compare with Meyer-Hall-Offin’s treatment?

Hill’s problem formulated in complex coordinates to abbreviate the form of the differential equation (cries out for computer algebra which of course Hill resp. Siegel didn’t have at his disposal when he did it). Proof by induction on terms in power series that it is possible. Again, convergence is proved by majorizing (note, Hill himself did not prove convergence, this was not done until Wintner in 1925).

Generalize the continuation method to the actual planar three-body problem (Meyer-Hall-Offin may give the impression that everything is relatively easy, not so here). One might be familiar with simple cases of solution by undetermined coefficients in power series from Coddington (our review here) or quantum-mechanics textbooks (our review here), the same idea invoked here but much more complicated! Abstraction of the continuation method for general systems of (analytic) ordinary differential equations. A fixed-point method due to Poincaré (use section, in an annulus with counterrotating rims there must by Birkhoff be at least two fixed points of the section mapping, see pp. 174-182).

Area-preserving transformation (as an abstraction of Poincaré’s section). Siegel and Moser will find a certain normal form (usually one stays in the category of Hamiltonian systems so one doesn’t need this). From power series expansion again, equation (23.15) invoking area-preserving property on p. 160. Convergence is easy in hyperbolic case, quite hard to show in elliptic due to small divisors. Lastly, Birkhoff’s fixed-point theorem and its application to Hamiltonian systems.

Chapter three is devoted to the technically demanding problem of showing stability in the near-integrable case. Note, instability is not defined as the logical negation of stability, allowing for possibility of mixed mappings. Schröder series found by convergence from majorization as usual (easier if λ does not have absolute value 1, finer estimates needed if it does but still direct using continued fractions and iteration, Cauchy formula; have to show domain of definition does not shrink to a point). Solve the partial differential equation to put into normal form by power series. A generalization of Poincaré’s center problem to more than one degree of freedom due to Liapunov. Assumption as to simplicity of eigenvalues dropped, normal form has to be suitably extended. A normal form for Hamiltonian systems: the problem is that the transformation into normal form might be divergent, nevertheless it can be used to approximate the solution (see statement on p. 216). See example constructed on pp. 217-218. There is no known finite method for deciding whether the transformation to normal form converges or diverges!

Finally, the KAM theorem in all its glory: the invariant torus is found by power series solution yielding a formal Fourier series the convergence of which is established by irrationality conditions on the periods – proof in p. 230ff as in Moser (reviewed by us here). The proof of the lemma is not very illuminating. Cauchy’s estimate to bound correction to linearization. This example shows clearly that the question of convergence does not depend on the number-theoretic properties of the eigenvalues, but rather on the nature of the non-linear terms – the opposite of what was the case for a conformal mapping in section 25. Stability of equilibrium solution to two degree of freedom Hamiltonian system reduced to theorem of section 33 by putting the differential equation into appropriate form, applied to L1-5 (maybe Meyer-Hall-Offin less compressed and so easier to follow).

Case of many degrees of freedom: Cauchy’s theorem allows one to shift the path of integration of Fourier coefficient to give it an imaginary part causing exponential decay of the Fourier coefficient = subclass of Bohr’s almost periodic functions particularly well adapted to problems of celestial mechanics. A formal power series expansion for the generating function and its application to the restricted three-body problem in Delaunay variables (NB, there are no analogous stability theorems for more than two degrees of freedom, a phenomenon known as Arnold diffusion). Poincaré’s recurrence theorem, applied to the restricted three-body problem on pp. 279-280 (not usually done!).

To summarize: although clear, the language has an old-fashioned ring to it compared say to Meyer-Hall-Offin (to be reviewed by us in a moment). Throughout these lectures, one will find very explicit formulae and constant employment of the tools of elementary real and complex analysis. The result proves to be pleasing in that although the formulae themselves can be intricate, the methods behind them are direct and concrete, undistracted by all the fancy technology besetting one elsewhere, such as Lie series.

NB, Siegel and Moser do not pursue perturbation theory in the usual sense of heuristic celestial mechanics, only a KAM-type approach in the last chapter. As a treatise on celestial mechanics, the present work leaves a lot to be desired, it is really a textbook in the theory of dynamical systems with special emphasis on Hamiltonian case and the N-body problem. For more state-of-the-art technology including coverage of classical perturbation theory, see Dmitry Treschev and Oleg Zubelevich’s Introduction to the Perturbation Theory of Hamiltonian Systems (Springer, 2015, which we intend to review in a moment as well).

A major drawback to these lectures by Siegel and Moser, however, is the absence of homework exercises. Also, there are no diagrams anywhere accompanying the text! Maybe reading Lichtenberg and Lieberman’s Regular and Chaotic Dynamics (Springer, second and expanded edition, 1992) would help in this respect? Despite fact that Siegel and Moser is the only place to get full proofs of many of the theorems referred to in other sources, reading it makes for hard going as one comes up against a profusion of statements, each singly not requiring more than basic real analysis but, as a whole, the presentation is unorganized. Still for the depth of the results it does contain, five stars.

Concluding reflection on the kind of science this book represents: not by any means revolutionary, in that by Siegel and Moser’s time celestial mechanics had grown to be a quite mature field, Poincaré’s stunning innovations were becoming normalized and tamed, so to speak. This evolution of the state of the art illustrates the to-and-fro of the progress of knowledge, for even in Poincaré’s time celestial mechanics had become highly developed compared to state in which Newton and Clairaut left it (due to ongoing contributions by d’Alembert, Lagrange, Laplace, Gauss, Jacobi, Newcomb, Weierstrass, Delaunay, Hill, Lindstedt and so forth) but, even so, Poincaré, in an exhibition of his genius, could revitalize the field with his novel perspectives. In the mid-twentieth century, KAM theory and allied topics represent a satisfying crowning achievement. Now that the revolution initiated by Poincaré is more or less complete, can one contemplate another similar revolution? Clearly the field needs to be infused once more with qualitatively new questions and approaches, but who knows, human ingenuity is restless and harbors an innate drive to the infinite, so maybe!

For instance, the question of stability in celestial mechanics can be tied to the problem of scarring in many-body quantum-mechanical systems, a current hot topic. So perhaps the investigation of classical KAM-type stability results in the context of quantum chaos could be an instance of what we have in mind – certainly there is plenty of room for the application of new techniques, the powerful methods of microlocal analysis have scarcely been exhausted in this domain.