What do you think?
Rate this book
Astronomy and Astrophysics Library
Theory of Orbits: Volume 1: Integrable Systems and Non-perturbative Methods
Half a century ago, S. Chandrasekhar wrote these words in the preface to his l celebrated and successful In this monograph an attempt has been made to present the theory of stellar dy namics as a branch of classical dynamics - a discipline in the same general category as celestial mechanics. [ ... J Indeed, several of the problems of modern stellar dy namical theory are so severely classical that it is difficult to believe that they are not already discussed, for example, in Jacobi's Vorlesungen. Since then, stellar dynamics has developed in several directions and at var ious levels, basically three viewpoints remaining from which to look at the problems encountered in the interpretation of the phenomenology. Roughly speaking, we can say that a stellar system (cluster, galaxy, etc.) can be con sidered from the point of view of celestial mechanics (the N-body problem with N » 1), fluid mechanics (the system is represented by a material con tinuum), or statistical mechanics (one defines a distribution function for the positions and the states of motion of the components of the system).
405 pages, Hardcover
First published March 1, 1996
9 people want to read
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 of 1 review
October 26, 2023
A refreshing take on a classical if not altogether dusty subject, that of celestial mechanics! In their Theory of Orbits (Spinger-Verlag Astronomy & Astrophysics Library, corrected third printing 2004), Dino Boccaletti and Giuseppe Pucacco from the University of Rome (departments of mathematics resp. physics) strive to unite celestial mechanics and stellar dynamics. Formally, these both represent instances of the gravitational N-body problem, with either just a few bodies on the one hand or, on the other hand, very many (at least on the order of 10,000 in a globular cluster).
Prima facie, one might think this forced, as if it is done just because both are fields of astronomy. The astronomical problems are very distinct; what they can have in common is the analytical technique applied to solve them. Actually there is a deeper connection between the two when viewed in terms of the actions which are the appropriate tool in terms of which to identify and integrate out short-period terms: see vol. 2, chapter nine on the theory of adiabatic invariants. For, in the case of the solar system, nobody would seriously contemplate integrating the straight Newtonian equation of motion with the universal law of gravitation in Cartesian coordinates. Rather, one wants to go to the Keplerian orbital elements and seek a development of the perturbative function into a power series. For, as long as the planets do not approach one another very closely, the orbital elements are slowly changing and one can adopt a longer time-step without sacrificing numerical precision. In the same vein, in a stellar dynamical system one wants to consider the distribution function as depending on action integrals rather than directly on the Cartesian coordinates.
Already early on in reading this text one encounters what is distinctive about the authors’ style, a preference for very computational proofs which need not, in themselves, be all that impressive, indeed can be hard to follow, for someone raised on Thirring or Arnold. One appreciates, nevertheless, that they do carry out explicit evaluations of the quantities sought – for instance, of the action-angle variables for the pendulum or for the harmonic oscillator. After an extensive section on the two-body problem (including introduction of Delaunay’s elements [pp. 158-162]), Boccaletti and Pucacco take up the problem of regularization [pp. 162-167]: what is interesting, and proves to be less hard that one might expect, if one appeals to the Levi-Civita transformation.
The subject of chapter three is the full N-body problem and what little one can show about it qualitatively from very general considerations. The first major result is analyticity of the solution up to a certain time [p. 183]. Typically, reference is made to Siegel and Moser for proof! Two general results on the N-body illustrate well Boccaletti and Pucacco’s direct computationally-oriented approach: first, they derive the standard quantities (corresponding to conservation of linear and angular momentum) by looking for integrals in an unsystematic way: by a hunch form a quantity which when substituted into the equation of motion becomes a total derivative [p. 185]. The nature of isolating integrals will be explained in section 5.1: Jacobi’s elimination of nodes reduces order by two more (not really explained here). The other, more systematic approach would be by direct application of Noether’s theorem to give ultimately five integrals in involution [pp. 188-192]! It is nice to see Noether’s theorem, often quoted in the abstract, put to practical use!
Painlevé and von Zeipel’s result on singularities are discussed but not proved [p. 194]. Though the collision set is of measure zero there’s no way to identify it, except for the case of a global collapse. Incidental remark: the recent work on the problem looks rather too technical (symptomatic of late-stage science). Sundman’s theorem is not after all difficult: use the Cauchy-Schwartz inequality on the expression for dI/dt (this was silently a homework problem in Thirring). A few interesting results on null energy, positive nergy and escape are quoted but proofs have to be omitted as too hard (relying on the Tauberian theorem from real analysis). A careful proof of the all-important virial theorem follows [pp. 210-211].
In this reviewer’s opinion, the particular solutions of N-body problem covered next in chapter three [pp. 216-235] are to be viewed not as an intrinisically interesting subject, but just as a curiosity. However that may be, the analysis, though a little involved, is not impossibly much so. At least one gets afterward a somewhat interesting analysis of Lagrangian solutions to the three-body problem.
In chapter four one specializes to N = 3. The problem is classic since, by counting degrees of freedom, one finds for three bodies that there are not enough integrals in involution to reduce it to integrability yet not so many degrees of freedom as to render it wholly intractable (especially if one goes to the restricted case where one of the bodies is infinitesimal compared to the other two).
Sundman’s regularization in 1d is sketched [pp. 246-248] and the Levi-Civita regularization explained in detail [pp. 248-256]. The significance of Jacobi’s integral in the restricted 3-body problem is that the path to the usual integrals is blocked by the assumption of vanishing of the third body’s mass. So it is fortunate that one still has an integral around to fall back upon! Tisserand’s criterion is easy to derive from Jacobi’s integral once the relevant approximations are enforced [p. 260]. Proceeding to a stability analysis of the Lagrange equilibria, the authors point out why stability of equilibrium solutions is here a non-trivial issue: the approximations made disrupt the applicability of the usual stability criterion [p. 266]. Then pp. 266-271 race through the mathematics without much insight (for instance, they appeal without explanation to Birkhoff normal form; for this, see Meyer-Hall-Offin). But, as a reward for the reader’s persistence, Delaunay elements for the restricted three-body problem are introduced and one winds up with a nice explicit expression for the canonical transformation [p. 276].
Extensions of the restricted three-body problem [pp. 284-299] are both conceptually interesting and relevant to actual astronomical problems in the solar system – unlike what is the case with special solutions to the N-body problem in chapter three. NB, in the planar elliptic restricted problem the Jacobi integral no longer exists! But Lagrangian solutions still do, leading to Sitnikov’s problem (for detail on this, see our review of Moser here).
The final topic in chapter three is Hill’s lunar problem [pp. 293-299], which may be envisioned as representing the sun-earth-moon system as an extension of the restricted problem when the sun is treated as infinitely far away but also infinitely massive so that a limiting form of its effect on the moon can be extracted. Here, after setting up the framework Boccaletti and Pucacco merely quote Hill’s result of regularization via Levi-Civita method [p. 295].
The last chapter five bears comparison with similar material in Binney-Tremaine, Galactic Dynamics (Princeton University Press, second edition, 2008). Isochronal potentials are a tour de force: first, a deformation of the Keplerian potential, then they find action variables in explicit form (as extensions of Delaunay elements). From p. 324ff, a nice discussion of ellipsoidal coordinates (in both directions). The problem of two fixed centers of attraction [pp. 341-348] may be compared with Thirring’s exemplary treatment in the first volume of his course in mathematical physics (see our review here). The last topic considered in vol. 1 is that of orbits in a triaxial potential, relevant to elliptical galaxies – which leads to nice figures since the coordinates have been chosen to be separable by Stäckel’s theorem. But unlike Binney and Tremaine, Boccaletti and Pucacco include no discussion of Martin Schwarzschild’s original method of building up a self-consistent model of an elliptical galaxies, in which families of orbits are populated in such a way that the gravitational potential obtained by integration of the Poisson equation coincides with that used to generate the orbits in the first place!
Overall, despite a consistently workmanlike level of exposition, the absence of homework exercises constitutes a serious drawback. This reviewer strongly recommends the more mathematically inclined Kenneth R. Meyer, Glen R. Hall and Dan Offin’s Introduction to Hamiltonian Dynamics and the N-Body Problem (Springer, Applied Mathematical Sciences vol. 9, 2009, which we intend to review in a moment) as a better place to learn some of the material for real by diligently working the exercises (or maybe Ferdinand Verhulst’s Nonlinear Differential Equations and Dynamical Systems, Springer Universitext, 2000). For the rest, go to Siegel-Moser (which we shall review momentarily) then the primary literature or specialized monographs (such as Szebehely) – in an extensive and time-honored field such as this, there are a lot of options with which this reviewer is little familiar, consult the pretty extensive bibliographical notes. Nonetheless, the present work by Boccaletti and Pucacco is worth a read-through and keeping around as a reference, for the authors do occasionally enliven their exposition with comments of a pedagogical nature, as to why they choose to do things their way or as to the significance of what is being done (for instance, they helpfully point out at one place that the secular terms appearing in classical perturbative expansions are an artifact of the method of solution and do not indicate anything as to the nature of the physical problem).
Prima facie, one might think this forced, as if it is done just because both are fields of astronomy. The astronomical problems are very distinct; what they can have in common is the analytical technique applied to solve them. Actually there is a deeper connection between the two when viewed in terms of the actions which are the appropriate tool in terms of which to identify and integrate out short-period terms: see vol. 2, chapter nine on the theory of adiabatic invariants. For, in the case of the solar system, nobody would seriously contemplate integrating the straight Newtonian equation of motion with the universal law of gravitation in Cartesian coordinates. Rather, one wants to go to the Keplerian orbital elements and seek a development of the perturbative function into a power series. For, as long as the planets do not approach one another very closely, the orbital elements are slowly changing and one can adopt a longer time-step without sacrificing numerical precision. In the same vein, in a stellar dynamical system one wants to consider the distribution function as depending on action integrals rather than directly on the Cartesian coordinates.
Already early on in reading this text one encounters what is distinctive about the authors’ style, a preference for very computational proofs which need not, in themselves, be all that impressive, indeed can be hard to follow, for someone raised on Thirring or Arnold. One appreciates, nevertheless, that they do carry out explicit evaluations of the quantities sought – for instance, of the action-angle variables for the pendulum or for the harmonic oscillator. After an extensive section on the two-body problem (including introduction of Delaunay’s elements [pp. 158-162]), Boccaletti and Pucacco take up the problem of regularization [pp. 162-167]: what is interesting, and proves to be less hard that one might expect, if one appeals to the Levi-Civita transformation.
The subject of chapter three is the full N-body problem and what little one can show about it qualitatively from very general considerations. The first major result is analyticity of the solution up to a certain time [p. 183]. Typically, reference is made to Siegel and Moser for proof! Two general results on the N-body illustrate well Boccaletti and Pucacco’s direct computationally-oriented approach: first, they derive the standard quantities (corresponding to conservation of linear and angular momentum) by looking for integrals in an unsystematic way: by a hunch form a quantity which when substituted into the equation of motion becomes a total derivative [p. 185]. The nature of isolating integrals will be explained in section 5.1: Jacobi’s elimination of nodes reduces order by two more (not really explained here). The other, more systematic approach would be by direct application of Noether’s theorem to give ultimately five integrals in involution [pp. 188-192]! It is nice to see Noether’s theorem, often quoted in the abstract, put to practical use!
Painlevé and von Zeipel’s result on singularities are discussed but not proved [p. 194]. Though the collision set is of measure zero there’s no way to identify it, except for the case of a global collapse. Incidental remark: the recent work on the problem looks rather too technical (symptomatic of late-stage science). Sundman’s theorem is not after all difficult: use the Cauchy-Schwartz inequality on the expression for dI/dt (this was silently a homework problem in Thirring). A few interesting results on null energy, positive nergy and escape are quoted but proofs have to be omitted as too hard (relying on the Tauberian theorem from real analysis). A careful proof of the all-important virial theorem follows [pp. 210-211].
In this reviewer’s opinion, the particular solutions of N-body problem covered next in chapter three [pp. 216-235] are to be viewed not as an intrinisically interesting subject, but just as a curiosity. However that may be, the analysis, though a little involved, is not impossibly much so. At least one gets afterward a somewhat interesting analysis of Lagrangian solutions to the three-body problem.
In chapter four one specializes to N = 3. The problem is classic since, by counting degrees of freedom, one finds for three bodies that there are not enough integrals in involution to reduce it to integrability yet not so many degrees of freedom as to render it wholly intractable (especially if one goes to the restricted case where one of the bodies is infinitesimal compared to the other two).
Sundman’s regularization in 1d is sketched [pp. 246-248] and the Levi-Civita regularization explained in detail [pp. 248-256]. The significance of Jacobi’s integral in the restricted 3-body problem is that the path to the usual integrals is blocked by the assumption of vanishing of the third body’s mass. So it is fortunate that one still has an integral around to fall back upon! Tisserand’s criterion is easy to derive from Jacobi’s integral once the relevant approximations are enforced [p. 260]. Proceeding to a stability analysis of the Lagrange equilibria, the authors point out why stability of equilibrium solutions is here a non-trivial issue: the approximations made disrupt the applicability of the usual stability criterion [p. 266]. Then pp. 266-271 race through the mathematics without much insight (for instance, they appeal without explanation to Birkhoff normal form; for this, see Meyer-Hall-Offin). But, as a reward for the reader’s persistence, Delaunay elements for the restricted three-body problem are introduced and one winds up with a nice explicit expression for the canonical transformation [p. 276].
Extensions of the restricted three-body problem [pp. 284-299] are both conceptually interesting and relevant to actual astronomical problems in the solar system – unlike what is the case with special solutions to the N-body problem in chapter three. NB, in the planar elliptic restricted problem the Jacobi integral no longer exists! But Lagrangian solutions still do, leading to Sitnikov’s problem (for detail on this, see our review of Moser here).
The final topic in chapter three is Hill’s lunar problem [pp. 293-299], which may be envisioned as representing the sun-earth-moon system as an extension of the restricted problem when the sun is treated as infinitely far away but also infinitely massive so that a limiting form of its effect on the moon can be extracted. Here, after setting up the framework Boccaletti and Pucacco merely quote Hill’s result of regularization via Levi-Civita method [p. 295].
The last chapter five bears comparison with similar material in Binney-Tremaine, Galactic Dynamics (Princeton University Press, second edition, 2008). Isochronal potentials are a tour de force: first, a deformation of the Keplerian potential, then they find action variables in explicit form (as extensions of Delaunay elements). From p. 324ff, a nice discussion of ellipsoidal coordinates (in both directions). The problem of two fixed centers of attraction [pp. 341-348] may be compared with Thirring’s exemplary treatment in the first volume of his course in mathematical physics (see our review here). The last topic considered in vol. 1 is that of orbits in a triaxial potential, relevant to elliptical galaxies – which leads to nice figures since the coordinates have been chosen to be separable by Stäckel’s theorem. But unlike Binney and Tremaine, Boccaletti and Pucacco include no discussion of Martin Schwarzschild’s original method of building up a self-consistent model of an elliptical galaxies, in which families of orbits are populated in such a way that the gravitational potential obtained by integration of the Poisson equation coincides with that used to generate the orbits in the first place!
Overall, despite a consistently workmanlike level of exposition, the absence of homework exercises constitutes a serious drawback. This reviewer strongly recommends the more mathematically inclined Kenneth R. Meyer, Glen R. Hall and Dan Offin’s Introduction to Hamiltonian Dynamics and the N-Body Problem (Springer, Applied Mathematical Sciences vol. 9, 2009, which we intend to review in a moment) as a better place to learn some of the material for real by diligently working the exercises (or maybe Ferdinand Verhulst’s Nonlinear Differential Equations and Dynamical Systems, Springer Universitext, 2000). For the rest, go to Siegel-Moser (which we shall review momentarily) then the primary literature or specialized monographs (such as Szebehely) – in an extensive and time-honored field such as this, there are a lot of options with which this reviewer is little familiar, consult the pretty extensive bibliographical notes. Nonetheless, the present work by Boccaletti and Pucacco is worth a read-through and keeping around as a reference, for the authors do occasionally enliven their exposition with comments of a pedagogical nature, as to why they choose to do things their way or as to the significance of what is being done (for instance, they helpfully point out at one place that the secular terms appearing in classical perturbative expansions are an artifact of the method of solution and do not indicate anything as to the nature of the physical problem).
Displaying 1 of 1 review