Structure of Dynamical Systems: A Symplectic View of Physics

Description

Souriau coined the term "Lie group thermodynamics"; cf. Barbaresco 2014 or his 2017 lecture, which mentions Souriau ch. 4.

Is the translator, de Vries, the same one as the KdV equation? Answer: No. That De Vries was Gustav.

This book is the English translation of Souriau's influential work [Structure des systèmes dynamiques, Dunod, Paris, 1970; MR0260238]. The translation editors have added the subtitle "a symplectic view of physics'', which is close to the title first proposed by the author and also indicates more adequately the spirit of this work. The book presents a self-contained and rigorous treatment of the symplectic approach to classical, statistical and quantum mechanics, primarily addressed to graduate students and researchers interested in mathematical and theoretical physics.
The roots of symplectic geometry can be traced back to the work of the founding fathers of analytical mechanics in the 18th and 19th centuries, such as Lagrange, Poisson, Hamilton and Jacobi. It was not, however, till the 1950s that the term "symplectic structure'' made its appearance and symplectic geometry developed into a distinct field of research in mathematics. To a large extent it has been the continuing and fruitful interaction with classical mechanics which, by the end of the 1960s, then led to the final breakthrough of symplectic geometry. It is precisely at this point that Souriau's work enters the picture. His above-cited book indeed represents the first complete treatment of mechanics which makes full use of the language and techniques of modern symplectic geometry.
In the first chapter, the author presents some basic definitions and results from differential geometry (the theory of manifolds, vector fields, differential forms, foliations, Lie groups) and the calculus of variations. Chapter II deals with the properties of symplectic (and presymplectic) structures. Special attention is paid to the theory of dynamical groups of symplectic and presymplectic manifolds, and to the notion of moment of a dynamical group (nowadays called the moment or momentum map). Some aspects of the cohomology of Lie groups and Lie algebras are discussed, and a construction is given of the symplectic form on the orbits of the coadjoint action of a Lie group on the dual of its Lie algebra.
Chapter III is devoted to the symplectic treatment of classical mechanics, nonrelativistic as well as relativistic. The evolution space of a mechanical system is equipped with a distinguished 22-form, called the Lagrange form, which contains all the essential information about the system. Nonrelativistic mechanics is then governed by two basic principles, namely: the closedness of the Lagrange form, which Souriau refers to as Maxwell's principle, and the principle of Galilean relativity for isolated systems. The flow of a dynamical system induces a 11-dimensional foliation on the evolution space, namely the characteristic foliation of the Lagrange form. The corresponding quotient space, i.e. the space of motions, admits a natural symplectic structure and, in case of an isolated system, the Galilei group is a dynamical group of this symplectic manifold. The characteristic properties of the space of motion of a mechanical system are subsequently promoted to the basic axioms of nonrelativistic mechanics. A similar treatment then follows for relativistic systems, with a Minkowski metric being defined on 44-dimensional space-time, and with the Galilei group being replaced by the (restricted) Poincarégroup.
The symplectic treatment of mechanics in particular leads to a classification of elementary relativistic particles, with or without spin and with or without mass, in terms of the energy-momentum vector and the polarization vector deduced from the moment of the restricted Poincarégroup. The last section of Chapter III deals with various topics such as the study of a material point in an electromagnetic field, systems of interacting particles, scattering theory, geometric optics and the theory of collisions of free particles.
Chapter IV, which is perhaps the least known part of Souriau's book, is devoted to a symplectic formulation of statistical mechanics. It starts with a discussion of measure theory on manifolds. The concept of statistical state of a dynamical system is defined as a probability measure on the symplectic space of motions of the system, and to each such measure one can assign a notion of entropy. An ensemble of statistical states, generalizing the Gibbs canonical ensemble, is introduced by requiring the entropy to be maximal, modulo some conditions involving a dynamical group. The basic concepts and principles of classical thermodynamics and of the kinetic theory of gases are then recovered within the present setting. Associating Gibbs ensembles to various subgroups of the Poincaréand Galilei groups, the author succeeds in deriving some interesting (new) phenomena such as relativistic ideal gases, the rotation of celestial bodies and classical and relativistic centrifuges.
The last chapter (Chapter V) deals with the problem of quantization and presents the basic ideas of the theory of geometric quantization, of which Souriau is one of the founding fathers. The notion of prequantum manifold is introduced, and necessary and sufficient conditions are established for a given symplectic manifold to admit the construction of an associated prequantum manifold. The possible role of these prequantum manifolds in the formulation of quantum mechanics is then investigated. Some of the highlights of this chapter are the derivation of various wave equations (the Schrödinger, Klein-Gordon, Dirac and the quantized Maxwell equations), and the construction of the Fock manifold of an "assemblée'' of particles (i.e. a system of an indeterminate number of indistinguishable particles). In the general introduction, the author carefully points out that the quantization method he proposes remains conjectural.
In spite of the enormous activity that has been developed in the field of symplectic geometry and mechanics since 1970, Souriau's book has not lost any of its value. Both for the wealth of information it contains, as well as for the author's original and inspiring view on the structure of the basic theories of physics, this book may be strongly recommended to anyone interested in the subject. The translation of this masterpiece therefore certainly yields a most welcome supplement to the existing English literature in this domain. Compared to the French version, the translation editors have changed some of the notations in order to put them in better agreement with those more commonly used in the current literature (for instance, the exterior derivative operator is denoted by the symbol dd instead of ∇\nabla). In various footnotes they also provide some useful additional information, for instance, to point out progress that has been made on certain problems since the appearance of the French edition.

Reviewed by Frans Cantrijn