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Astronomy and Astrophysics Library
Theory of Orbits: Perturbative and Geometrical Methods
Theory of Orbits treats celestial mechanics as well as stellar dynamics from the common point of view of orbit theory, making use of concepts and techniques from modern geometric mechanics. It starts with elementary Newtonian mechanics and ends with the dynamics of chaotic motion. The two volumes are meant for students in astronomy and physics alike. Prerequisite is a physicist's knowledge of calculus and differential geometry.
437 pages, Hardcover
First published November 19, 1998
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June 4, 2024
See our companion review of the Theory of Orbits, vol. 1, Integrable Systems and Non-Perturbative Methods (here). In vol. 2, Perturbative and Geometrical Methods, Dino Boccaletti and Giuseppe Pucacco continue their thorough treatment of celestial mechanics in the straightforward, mathematically unpretentious style to which we have become accustomed in the first volume.
The primary aim of the second volume is to provide an in-depth coverage of classical perturbation theory. In chapter six, the basics – Gauss’ method for the secular terms, Poisson’s equations, variation of elements, Lindstedt’s device, the averaging method and a few other problems, such as motion around an oblate spheroid and Kepler’s problem with drag. In chapter seven, canonical perturbation theory; i.e., a perturbative method in which one ensures that the changes of coordinate will always be canonical transformations. Jacobi’s method shows how to set up the transformation equations, at least to first order, and is particularly meaningful when carried out in terms of Delaunay elements. Here, the Duffing equation serves as very simple proxy for Lindstedt-Poincaré method, exhibited in detail.
But one would like to be able to continue the perturbative expansion to higher order. To do so, a convenient method has been devised by H. von Zeipel in 1916, which consists in expanding the generating function of the canonical transformation as a power series in the small parameter. Equating coefficients of the same power yields the new Hamiltonian at each stage. The method is illustrated in detail in systems of one degree of freedom, many degrees of freedom and in twist maps. Here, the KAM theorem is discussed at length but without proof (the real analysis required for its proof goes well beyond the mathematical level assumed for the audience of this textbook). The destruction of invariant tori and its connection with arithmetic properties of the frequencies described in detail, following M.V. Berry (here, it helps to look at the continued fraction expansion).
An unfortunate feature of von Zeipel’s method is that is requires computations of the change of variable in terms of a generating function, which mixes the old and new (an amazing statistic: Delaunay in the nineteenth century computed as many as 505 successive canonical transformations!). It would be calculationally easier if one could map directly from the old to the new. The modern method of Lie transforms accomplishes just this! Originally due to G. Hori and A. Deprit in 1966, it allows for a near-identity transformation by formally exponentiating the Lie derivative. The application of Lie series to perturbation theory is known as the Lie transform method, for which Deprit invented a complicated procedure that speeds up convergence by eliminating as many terms as possible at each step. Boccaletti and Pucacco then discuss a generalization to not necessarily Hamiltonian systems due to Kamel and apply it to the Duffing equation, the pendulum, the parametrically driven non-linear oscillator and a perturbed isochronal potential. Two more sections in chapter eight address Hamiltonian normal forms and Kolmogorov’s superconvergent series technique.
Adiabatic invariants – made famous by Einstein in 1911 who pointed out their relevance to the old quantum theory – form the subject of chapter nine. Boccaletti and Puccaco provide a formal definition of an adiabatic invariant – the author are always good on defining what they mean by their terms in clear, modern notation – and show adiabatic invariance of the action directly without any asymptotic expansion. Then they give a nice intuitive exemplification of the ideas with a charged particle in an electromagnetic field, based on separation of scales.
Chapter ten discusses period orbits and resonances in celestial mechanics, which provide an excuse to go into bifurcation theory. Here, the authors often refer to Siegel-Moser or Meyer-Hall-Offin. In section 10.5, these ideas are supplied with some relevance by means of an analysis of the bifurcation theory in the case of the L4 and L5 Lagrange points in the restricted three-body problem. From section 10.6 (global results) onwards the treatment becomes increasingly sketchy. Chapter eleven on chaos seems to be somewhat loosely connected with the subject matter of the previous chapters (maybe Lichtenberg and Lieberman would be a better and more comprehensive reference?). Likewise for chapter twelve dedicated solely to numerical methods. Apart from section 12.4.3 (chaos in the solar system, a summary of Laskar’s work), it isn’t clear that the material belongs in this book.
At least for chapters six through ten, a highly readable account of modern perturbative methods in celestial mechanics. A good point about Boccaletti and Puccaco is that, throughout this and the previous volume, they do like to state the criteria in clear mathematical terminology, as in their clear discussion of Krylov-Bogoliubov method in chapter six – even though the style of reasoning they employ is closer to heuristic theoretical physics than to rigorous mathematical physics, the use everywhere of clear mathematical definitions of the terms and statement of the theorems will certainly facilitate the reader’s transition to the more mathematical literature, should he elect to pursue it. Another good point of the exposition is the level of detail, in particular in the examples, the results of which are expressed in explicit formulae wherever possible.
Therefore, we can award a higher rating to volume two of the Theory of Orbits than we did to volume one: four stars! Recommended to the student who wishes to see the formalism not merely elucidated in the abstract, but worked out in detail in examples.
The primary aim of the second volume is to provide an in-depth coverage of classical perturbation theory. In chapter six, the basics – Gauss’ method for the secular terms, Poisson’s equations, variation of elements, Lindstedt’s device, the averaging method and a few other problems, such as motion around an oblate spheroid and Kepler’s problem with drag. In chapter seven, canonical perturbation theory; i.e., a perturbative method in which one ensures that the changes of coordinate will always be canonical transformations. Jacobi’s method shows how to set up the transformation equations, at least to first order, and is particularly meaningful when carried out in terms of Delaunay elements. Here, the Duffing equation serves as very simple proxy for Lindstedt-Poincaré method, exhibited in detail.
But one would like to be able to continue the perturbative expansion to higher order. To do so, a convenient method has been devised by H. von Zeipel in 1916, which consists in expanding the generating function of the canonical transformation as a power series in the small parameter. Equating coefficients of the same power yields the new Hamiltonian at each stage. The method is illustrated in detail in systems of one degree of freedom, many degrees of freedom and in twist maps. Here, the KAM theorem is discussed at length but without proof (the real analysis required for its proof goes well beyond the mathematical level assumed for the audience of this textbook). The destruction of invariant tori and its connection with arithmetic properties of the frequencies described in detail, following M.V. Berry (here, it helps to look at the continued fraction expansion).
An unfortunate feature of von Zeipel’s method is that is requires computations of the change of variable in terms of a generating function, which mixes the old and new (an amazing statistic: Delaunay in the nineteenth century computed as many as 505 successive canonical transformations!). It would be calculationally easier if one could map directly from the old to the new. The modern method of Lie transforms accomplishes just this! Originally due to G. Hori and A. Deprit in 1966, it allows for a near-identity transformation by formally exponentiating the Lie derivative. The application of Lie series to perturbation theory is known as the Lie transform method, for which Deprit invented a complicated procedure that speeds up convergence by eliminating as many terms as possible at each step. Boccaletti and Pucacco then discuss a generalization to not necessarily Hamiltonian systems due to Kamel and apply it to the Duffing equation, the pendulum, the parametrically driven non-linear oscillator and a perturbed isochronal potential. Two more sections in chapter eight address Hamiltonian normal forms and Kolmogorov’s superconvergent series technique.
Adiabatic invariants – made famous by Einstein in 1911 who pointed out their relevance to the old quantum theory – form the subject of chapter nine. Boccaletti and Puccaco provide a formal definition of an adiabatic invariant – the author are always good on defining what they mean by their terms in clear, modern notation – and show adiabatic invariance of the action directly without any asymptotic expansion. Then they give a nice intuitive exemplification of the ideas with a charged particle in an electromagnetic field, based on separation of scales.
Chapter ten discusses period orbits and resonances in celestial mechanics, which provide an excuse to go into bifurcation theory. Here, the authors often refer to Siegel-Moser or Meyer-Hall-Offin. In section 10.5, these ideas are supplied with some relevance by means of an analysis of the bifurcation theory in the case of the L4 and L5 Lagrange points in the restricted three-body problem. From section 10.6 (global results) onwards the treatment becomes increasingly sketchy. Chapter eleven on chaos seems to be somewhat loosely connected with the subject matter of the previous chapters (maybe Lichtenberg and Lieberman would be a better and more comprehensive reference?). Likewise for chapter twelve dedicated solely to numerical methods. Apart from section 12.4.3 (chaos in the solar system, a summary of Laskar’s work), it isn’t clear that the material belongs in this book.
At least for chapters six through ten, a highly readable account of modern perturbative methods in celestial mechanics. A good point about Boccaletti and Puccaco is that, throughout this and the previous volume, they do like to state the criteria in clear mathematical terminology, as in their clear discussion of Krylov-Bogoliubov method in chapter six – even though the style of reasoning they employ is closer to heuristic theoretical physics than to rigorous mathematical physics, the use everywhere of clear mathematical definitions of the terms and statement of the theorems will certainly facilitate the reader’s transition to the more mathematical literature, should he elect to pursue it. Another good point of the exposition is the level of detail, in particular in the examples, the results of which are expressed in explicit formulae wherever possible.
Therefore, we can award a higher rating to volume two of the Theory of Orbits than we did to volume one: four stars! Recommended to the student who wishes to see the formalism not merely elucidated in the abstract, but worked out in detail in examples.
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