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Celestial Encounters
Celestial Encounters is for anyone who has ever wondered about the foundations of chaos. In 1888, the 34-year-old Henri Poincaré submitted a paper that was to change the course of science, but not before it underwent significant changes itself. "The Three-Body Problem and the Equations of Dynamics" won a prize sponsored by King Oscar II of Sweden and Norway and the journal Acta Mathematica , but after accepting the prize, Poincaré found a serious mistake in his work. While correcting it, he discovered the phenomenon of chaos.
Starting with the story of Poincaré's work, Florin Diacu and Philip Holmes trace the history of attempts to solve the problems of celestial mechanics first posed in Isaac Newton's Principia in 1686. In describing how mathematical rigor was brought to bear on one of our oldest fascinations--the motions of the heavens--they introduce the people whose ideas led to the flourishing field now called nonlinear dynamics.
In presenting the modern theory of dynamical systems, the models underlying much of modern science are described pictorially, using the geometrical language invented by Poincaré. More generally, the authors reflect on mathematical creativity and the roles that chance encounters, politics, and circumstance play in it.
Starting with the story of Poincaré's work, Florin Diacu and Philip Holmes trace the history of attempts to solve the problems of celestial mechanics first posed in Isaac Newton's Principia in 1686. In describing how mathematical rigor was brought to bear on one of our oldest fascinations--the motions of the heavens--they introduce the people whose ideas led to the flourishing field now called nonlinear dynamics.
In presenting the modern theory of dynamical systems, the models underlying much of modern science are described pictorially, using the geometrical language invented by Poincaré. More generally, the authors reflect on mathematical creativity and the roles that chance encounters, politics, and circumstance play in it.
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First published November 11, 1996
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December 1, 2023
The venerable field of celestial mechanics appeals to mankind’s deep-going impression of the place of the heavens in the created world. For the ancients, the heavens revolve as a realm of stately order in stark contrast to the mutability of the sublunary region – what, for instance, the psalmist must have had in the back of his mind when he writes,
The heavens declare the glory of God,
the vault of heaven proclaims his handiwork,
day discourses of it to day,
night to night hands on the knowledge. (Psalm 19:1-2)
But after the demotion of Aristotelian cosmology that took place during the early modern period, what becomes of the heavens’ glory? For sure, it grows increasingly abstract, if also thereby truer. Rather than an immediately perceptible reality, it becomes, if anything, an inference from experience. Why do we contend that it also truer than before? For the expansion of scientific knowledge means that we accede to a position of greater appreciation of what it is about, namely, its being a testament to God’s constancy and fidelity as expressed through the existence of dependable laws of nature.
Nicholas of Cusa represents an interesting way-station along the path to the modern world-view. Neither medieval nor a Copernican, Nicholas revolutionizes our view of the cosmos by elevating the earth to the company of the stars. With the abandonment of the ancient and medieval model of a spherical world centered on the earth, the age-old contrast between the heavens and the earth loses its force. What assumes its place, for Nicholas, is conceptual construction that mediates between the ideal divine realm and worldly realities – not, be it noted, a mathematical physics along the lines envisioned by Descartes, but, hearkening back to the medieval symbolist mentality, a hierarchical world of symbolic correspondences most fittingly appreciated in the language of higher mathematics (for our review of Nicholas of Cusa’s magnum opus, On learned ignorance, see here).
Alexander Pope’s epigram –
Nature and nature's laws lay hid in night; God said ‘Let Newton be’ and all was light
– is after all justified. At the hands of Newton, Clairaut, Laplace etc. celestial mechanics attains the status of an exact science that demonstrates, more than anything else until comparatively recently, that nature indeed complies with universal laws expressible in strictly mathematical form, a state of affairs to which we have been accustomed so long that we tend to lose sight of its strikingness. The so-called scientific revolution of the seventeenth century owes a substantial measure of its impress on modern culture to the successes of celestial mechanics alone, if one recall that at the time physics and chemistry were hardly quantitative – a status to which they attained only later in the nineteenth century.
Considerations such as these ought to motivate in the educated layman a desire to know more about the most up-to-date views on subjects surrounding celestial mechanics. A quirk of terminology not unlike the way in which the general theory of relativity might suggest, to the half-learned, a stance favoring relativism in the philosophical sense, means that we now refer to practitioners of the field as chaos theorists, as if all order has been disrupted and unpredictability unleashed. In reality, scientific knowledge remains as firm as ever. This reviewer highly recommends Florin Diacu and Philip Holmes’ Celestial Encounters: The Origins of Chaos and Stability (Princeton University Press, 1996) as a trustworthy guide that aims for more of a mathematical foundation than a number of other popular works (for instance, James Gleick’s Chaos – Making a New Science, 2011). The authors’ decision to focus on the gravitational N-body problem helps to cut down on what might otherwise threaten to be an unmanageable scope. Newton solves the two-body problem exactly and thereby derives Kepler’s three laws of planetary motion, but, as he is well aware, his universal law of gravitation renders the problem of more than two bodies in interaction extremely difficult though of central importance, if Newtonian classical mechanics is to connect with the real world as instantiated by the solar system. Many of the brightest theorists since Newton’s day have expended their efforts on it and made considerable progress on the N-body problem, despite its being, in the end, intractable – i.e., not just hard but outright unsolvable, if one wants a solution in closed analytical form; Sundman has shown that a solution given by convergent power series exists in principle yet in practice, the rate of convergence is hopelessly slow and the problem continues to be computationally challenging to get very far with it.
Daicu and Holmes tell the story of what we have come to know, and to know we cannot know, with panache. The material is standard, so what is of interest here is the route through the territory that the authors choose to trace. In a preliminary tour in chapter one, they opt for a gentle approach into their subject by way of telling the story of the great late-nineteenth-century French mathematician, Henri Poincaré. Apart from his monumental contributions to all areas of the mathematics of his time – in the theory of ordinary differential equations, algebraic topology and so forth – Poincaré engages their attention on account of his memoir that won the prize announced by king Oscar II of Sweden and Norway in 1885 and awarded in 1889, on the three-body problem.
Here, in lectures eventually published in 1893, Poincaré revolutionizes the field of celestial mechanics, despite failing to solve the problem per se (in so far as it has ever been ‘solved’, the solution was achieved by Sundman in 1907-1909). For he has an entirely new way of looking at the problem: rather than seek ever-more refined quantitative techniques by which to integrate the motion for given initial conditions – which by then, after Delaunay, had become quite advanced and, indeed, had made possible the discovery of the planet Neptune in 1846 – Poincaré invites us to turn our attention to the qualititative behavior of solutions as they range across the whole scope of potential scenarios. The so-called phase-space portraits with which we are so familiar today are conceived for the first time. Thus, Poincaré’s groundbreaking work affords Diacu and Holmes an excellent opportunity to set forth the elements of dynamical systems theory as we have learned from the master to view it – which it is their principal aim to convey to a lay audience in this book.
Chapter two launches into a seeming detour into what now gets called symbolic dynamics: rather than keep track of the precise position and velocity of all bodies at every given time, one satisfies oneself with recording the places they visit at a coursed-grained level; that is, attaches a unique symbol to each region in phase-space. The itinerary of the regions traversed by the system, presented as a sequence of symbols, is all the information one preserves. This approach turns out to be quite powerful, as it obviates the distraction of trying to maintain precise quantities and allows one’s focus to turn, instead, to the general features of the system’s behavior. Clearly, if the behavior is regular or stable, only a comparative few places will be visited over time in a repeating pattern of symbols, but if chaotic, the symbol sequence becomes correspondingly all but random. Thus, symbolic dynamics makes for an interesting starting point, not the one followed by the historical evolution of the discipline of classical mechanics (Galileo-Kepler-Newton) – presumably chosen as it renders the subject more tangible, allowing a fast-paced excursion to get to chaos proper which is harder to imagine for a continuous system.
Chapter three turns to an exciting topic pertaining to the N-body problem – collisions and other singularities. For, if the bodies be idealized as point masses, then they can approach one another arbitrarily close and the gravitational accelerations can increase without limit. Daicu and Holmes retrace the lines of development of work on the subject going back to a celebrated conjecture by a contemporary of Poincaré’s, Paul Painlevé which was followed up on around the turn of the twentieth century by Edvard Hugo von Zeipel. But what adds piquancy is the fact that it has not been exhausted by the classical results and, instead, continues to inspire a good deal of research down to the present day. Beginning in 1966, the problem attracted the attention of Victor Szebehly, John Mather and most recently Jeff Xia, who has shown the existence of complicated recurring patterns of motion executed by any number of bodies – the so-called ‘choreographies’.
The last two chapters concern the problem of when stability can be shown – that is, if we take a problem known to be integrable (such as the revolution of the planets around the sun when the gravitational forces among the planets themselves can be ignored), what happens when one turns on the perturbation? Is the stable, integrable motion preserved (up to the perturbation) or does the character of the solutions change completely, becoming unpredictable and chaotic? Either possibility can be realized, depending on how strong a perturbation is permitted.
The theorem collectively due to Kolmogorov, Arnold and Moser (Kolmogorov announced the result in 1954 to the International Congress of Mathematicians but never published a sufficient proof) confirms what intuition suggests, that in the limit when the perturbation parameter is taken to be vanishingly small (though non-zero) the integrable character of the solutions persists – for the most part, that is, or almost everywhere. The result is rather subtle, though, in that even for an arbitrarily small but non-zero perturbation parameter, the qualitative features of the phase-space portrait are entirely disrupted (a reflection of the fact that the power series solutions no longer converge everywhere due to the presence of small divisors). Diacu and Holmes’ coverage of the issues surrounding the problem of stability, extending all the way from Lagrange and Laplace in the eighteenth century up to current work on Arnold diffusion, is very inviting.
Does the present work by Diacu and Holmes live up to its promise to induct the layman into the mysteries of chaos theory? Yes, by and large. A technical point: though – as the title indicates – there is a stress on celestial mechanics proper, the book is really a propaedeutic to the modern theory of dynamical systems in general (where the gravitational N-body problem continues to play an important motivating role). Keeping the needs of the non-expert in mind, the authors include hardly any equations, but provide plenty of phase-space diagrams.
On the human interest side, one will find portraits of the main players and, throughout, technical exposition of the science is interwoven with biographical episodes. It is always instructive to be aware of the aspect of scientific research as a human endeavor. The biographical material is not overdone, though (sometimes the history of science may be treated as a pretext for commenting on general societal issues and the mathematical results themselves are relegated to a subordinate role, but not here).
All around, readable, lively, informative, should appeal to budding scientists or to open-minded non-specialist adults looking to expand their horizons.
The heavens declare the glory of God,
the vault of heaven proclaims his handiwork,
day discourses of it to day,
night to night hands on the knowledge. (Psalm 19:1-2)
But after the demotion of Aristotelian cosmology that took place during the early modern period, what becomes of the heavens’ glory? For sure, it grows increasingly abstract, if also thereby truer. Rather than an immediately perceptible reality, it becomes, if anything, an inference from experience. Why do we contend that it also truer than before? For the expansion of scientific knowledge means that we accede to a position of greater appreciation of what it is about, namely, its being a testament to God’s constancy and fidelity as expressed through the existence of dependable laws of nature.
Nicholas of Cusa represents an interesting way-station along the path to the modern world-view. Neither medieval nor a Copernican, Nicholas revolutionizes our view of the cosmos by elevating the earth to the company of the stars. With the abandonment of the ancient and medieval model of a spherical world centered on the earth, the age-old contrast between the heavens and the earth loses its force. What assumes its place, for Nicholas, is conceptual construction that mediates between the ideal divine realm and worldly realities – not, be it noted, a mathematical physics along the lines envisioned by Descartes, but, hearkening back to the medieval symbolist mentality, a hierarchical world of symbolic correspondences most fittingly appreciated in the language of higher mathematics (for our review of Nicholas of Cusa’s magnum opus, On learned ignorance, see here).
Alexander Pope’s epigram –
Nature and nature's laws lay hid in night; God said ‘Let Newton be’ and all was light
– is after all justified. At the hands of Newton, Clairaut, Laplace etc. celestial mechanics attains the status of an exact science that demonstrates, more than anything else until comparatively recently, that nature indeed complies with universal laws expressible in strictly mathematical form, a state of affairs to which we have been accustomed so long that we tend to lose sight of its strikingness. The so-called scientific revolution of the seventeenth century owes a substantial measure of its impress on modern culture to the successes of celestial mechanics alone, if one recall that at the time physics and chemistry were hardly quantitative – a status to which they attained only later in the nineteenth century.
Considerations such as these ought to motivate in the educated layman a desire to know more about the most up-to-date views on subjects surrounding celestial mechanics. A quirk of terminology not unlike the way in which the general theory of relativity might suggest, to the half-learned, a stance favoring relativism in the philosophical sense, means that we now refer to practitioners of the field as chaos theorists, as if all order has been disrupted and unpredictability unleashed. In reality, scientific knowledge remains as firm as ever. This reviewer highly recommends Florin Diacu and Philip Holmes’ Celestial Encounters: The Origins of Chaos and Stability (Princeton University Press, 1996) as a trustworthy guide that aims for more of a mathematical foundation than a number of other popular works (for instance, James Gleick’s Chaos – Making a New Science, 2011). The authors’ decision to focus on the gravitational N-body problem helps to cut down on what might otherwise threaten to be an unmanageable scope. Newton solves the two-body problem exactly and thereby derives Kepler’s three laws of planetary motion, but, as he is well aware, his universal law of gravitation renders the problem of more than two bodies in interaction extremely difficult though of central importance, if Newtonian classical mechanics is to connect with the real world as instantiated by the solar system. Many of the brightest theorists since Newton’s day have expended their efforts on it and made considerable progress on the N-body problem, despite its being, in the end, intractable – i.e., not just hard but outright unsolvable, if one wants a solution in closed analytical form; Sundman has shown that a solution given by convergent power series exists in principle yet in practice, the rate of convergence is hopelessly slow and the problem continues to be computationally challenging to get very far with it.
Daicu and Holmes tell the story of what we have come to know, and to know we cannot know, with panache. The material is standard, so what is of interest here is the route through the territory that the authors choose to trace. In a preliminary tour in chapter one, they opt for a gentle approach into their subject by way of telling the story of the great late-nineteenth-century French mathematician, Henri Poincaré. Apart from his monumental contributions to all areas of the mathematics of his time – in the theory of ordinary differential equations, algebraic topology and so forth – Poincaré engages their attention on account of his memoir that won the prize announced by king Oscar II of Sweden and Norway in 1885 and awarded in 1889, on the three-body problem.
Here, in lectures eventually published in 1893, Poincaré revolutionizes the field of celestial mechanics, despite failing to solve the problem per se (in so far as it has ever been ‘solved’, the solution was achieved by Sundman in 1907-1909). For he has an entirely new way of looking at the problem: rather than seek ever-more refined quantitative techniques by which to integrate the motion for given initial conditions – which by then, after Delaunay, had become quite advanced and, indeed, had made possible the discovery of the planet Neptune in 1846 – Poincaré invites us to turn our attention to the qualititative behavior of solutions as they range across the whole scope of potential scenarios. The so-called phase-space portraits with which we are so familiar today are conceived for the first time. Thus, Poincaré’s groundbreaking work affords Diacu and Holmes an excellent opportunity to set forth the elements of dynamical systems theory as we have learned from the master to view it – which it is their principal aim to convey to a lay audience in this book.
Chapter two launches into a seeming detour into what now gets called symbolic dynamics: rather than keep track of the precise position and velocity of all bodies at every given time, one satisfies oneself with recording the places they visit at a coursed-grained level; that is, attaches a unique symbol to each region in phase-space. The itinerary of the regions traversed by the system, presented as a sequence of symbols, is all the information one preserves. This approach turns out to be quite powerful, as it obviates the distraction of trying to maintain precise quantities and allows one’s focus to turn, instead, to the general features of the system’s behavior. Clearly, if the behavior is regular or stable, only a comparative few places will be visited over time in a repeating pattern of symbols, but if chaotic, the symbol sequence becomes correspondingly all but random. Thus, symbolic dynamics makes for an interesting starting point, not the one followed by the historical evolution of the discipline of classical mechanics (Galileo-Kepler-Newton) – presumably chosen as it renders the subject more tangible, allowing a fast-paced excursion to get to chaos proper which is harder to imagine for a continuous system.
Chapter three turns to an exciting topic pertaining to the N-body problem – collisions and other singularities. For, if the bodies be idealized as point masses, then they can approach one another arbitrarily close and the gravitational accelerations can increase without limit. Daicu and Holmes retrace the lines of development of work on the subject going back to a celebrated conjecture by a contemporary of Poincaré’s, Paul Painlevé which was followed up on around the turn of the twentieth century by Edvard Hugo von Zeipel. But what adds piquancy is the fact that it has not been exhausted by the classical results and, instead, continues to inspire a good deal of research down to the present day. Beginning in 1966, the problem attracted the attention of Victor Szebehly, John Mather and most recently Jeff Xia, who has shown the existence of complicated recurring patterns of motion executed by any number of bodies – the so-called ‘choreographies’.
The last two chapters concern the problem of when stability can be shown – that is, if we take a problem known to be integrable (such as the revolution of the planets around the sun when the gravitational forces among the planets themselves can be ignored), what happens when one turns on the perturbation? Is the stable, integrable motion preserved (up to the perturbation) or does the character of the solutions change completely, becoming unpredictable and chaotic? Either possibility can be realized, depending on how strong a perturbation is permitted.
The theorem collectively due to Kolmogorov, Arnold and Moser (Kolmogorov announced the result in 1954 to the International Congress of Mathematicians but never published a sufficient proof) confirms what intuition suggests, that in the limit when the perturbation parameter is taken to be vanishingly small (though non-zero) the integrable character of the solutions persists – for the most part, that is, or almost everywhere. The result is rather subtle, though, in that even for an arbitrarily small but non-zero perturbation parameter, the qualitative features of the phase-space portrait are entirely disrupted (a reflection of the fact that the power series solutions no longer converge everywhere due to the presence of small divisors). Diacu and Holmes’ coverage of the issues surrounding the problem of stability, extending all the way from Lagrange and Laplace in the eighteenth century up to current work on Arnold diffusion, is very inviting.
Does the present work by Diacu and Holmes live up to its promise to induct the layman into the mysteries of chaos theory? Yes, by and large. A technical point: though – as the title indicates – there is a stress on celestial mechanics proper, the book is really a propaedeutic to the modern theory of dynamical systems in general (where the gravitational N-body problem continues to play an important motivating role). Keeping the needs of the non-expert in mind, the authors include hardly any equations, but provide plenty of phase-space diagrams.
On the human interest side, one will find portraits of the main players and, throughout, technical exposition of the science is interwoven with biographical episodes. It is always instructive to be aware of the aspect of scientific research as a human endeavor. The biographical material is not overdone, though (sometimes the history of science may be treated as a pretext for commenting on general societal issues and the mathematical results themselves are relegated to a subordinate role, but not here).
All around, readable, lively, informative, should appeal to budding scientists or to open-minded non-specialist adults looking to expand their horizons.
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