The Geometry of Celestial Mechanics

Book Preface

Celestial mechanics has attracted the interest of some of the greatest mathematical minds in history, from the ancient Greeks to the present day. Isaac Newton’s deduction of the universal law of gravitation (Newton, 1687) triggered enormous advances in mathematical astronomy, spearheaded by the mathematical giant Leonhard Euler (1707–1783). Other mathematicians who drove the development of celestial mechanics in the first half of the eighteenth century were Alexis Claude Clairaut (1713–1765) and Jean le Rond d’Alembert (1717–1783), see (Linton, 2004). In those days, the demarcation lines separating mathematics and physics from each other and from intellectual life in general had not yet been drawn. Indeed, d’Alembert may be more famous as the co-editor with Denis Diderot of the Encyclop´edie. During the Enlightenment, celestial mechanics was a subject discussed in the salons by writers, philosophers and intellectuals like Voltaire (1694–1778) and ´ Emilie du Chˆatelet (1706–1749).

The history of celestial mechanics continues with Joseph-Louis Lagrange (1736–1813), Pierre-Simon de Laplace (1749–1827) and William Rowan Hamilton (1805–1865), to name but three mathematicians whose contributions will be discussed at length in this text. Henri Poincar´e (1854–1912), perhaps the last universal mathematician, initiated the modern study of the three-body problem, together with large parts of the theory of dynamical systems and what is now known as symplectic geometry (Barrow-Green, 1997; Charpentier et al., 2010; McDuff and Salamon, 1998).

Yet this time-honoured subject seems to have all but vanished from the mathematical curricula of our universities. This is reflected in the available textbooks, which are either getting a bit long in the tooth, or are addressed to a fairly advanced and specialised audience. The Lectures on Celestial Mechanics by Siegel and Moser (1971), a classic in their own right, deal with Sundman’s work on the three-body problem in the wake of Poincar´e’s, and with questions about periodic solutions and stability, all at a rather mature level. Celestial mechanics as a key motivation for the study of dynamical systems is served well by (Moser and Zehnder, 2005) and (Meyer et al., 2009).

My personal interest in celestial mechanics stems from reading the paper (Albers et al., 2012), where the three-body problem is approached with methods from contact topology, my core area of expertise, see (Geiges, 2008). I should say ‘attempting to read’, for I quickly realised that I was ignorant of some of the most basic terminology in celestial mechanics.

In order to remedy this deplorable state of affairs – and to confute the inquisitor – I decided to teach a course on celestial mechanics, with (Pollard, 1966), (Danby, 1992) and (Ortega and Ure˜na, 2010) as my excellent guides. The latter textbook can be recommended even to readers whose grasp of Spanish is as rudimentary as mine.