Jerrold E. Marsden, Tudor S. Ratiu:

Introduction to Mechanics
and Symmetry

A Basic Exposition of Classical Mechanical Systems

Springer Verlag, New York, 1999
Texts in Applied Mathematics series

Second edition, 582 p.,
with 54 illustrations,
References pp. 519-551,
Index pp. 554-582

Following the first edition (1994) the authors revised the text significantly. From our point of view, the most important change is the rewriting and expanding of the chapter on the Lie groups. This volume can be read as a new publication rather than a revised edition of the previous.

At most treatments, mechanics was traditionally considered as an axiomatic system, what could be derived from the basic laws of nature established by Newton, and this deductive description was extended by some later introduced mathematical tools or physical principles. It is well known, that while 17^th century physics was based on laws of the nature, 18^th century physics followed a different path: they built a physics on the variational principles. Since variational principles sought for extremums, and this supposed the invariance (constancy) of at least one parameter, all variational principles were more-or-less symmetry principles. (Please note: if the variation of an integral is 0, that means, the integral is stationary under the fixed conditions, what is equivalent with an invariance.) Variational principles of physics are not a consequence of the laws of Newton. An independent description of physics can be built by their means (what of course, does not contradict to the Newtonian mechanics and incorporates that). As B. van Fraassen wrote (in Laws and Symmetry, Oxford Univ. Press, 1989, 395 p., ) "Laws do appear in this view - but only laws of models, basic principles of the theory, fundamental equations. Some principles are indeed deeper or more fundamental than others. Pre-eminent among these are the symmetries of the models, intimately connected with the conservation laws, but ubiquitous in their influence on theory construction." (p. 188.) "Symmetries of the model ... are ‘deeper’ because they tell us something beforehand about what the laws of coexistence and succession can look like. It is in the twentieth century’s quantum theory, that symmetry, coexistence and succession became most elegantly joined and intricately connected." (p. 223.)

This program is executed and presented by Marsden and Ratiu in this book. As a consequence, the notion of "force" does not play a central role in this book on mechanics, like in most traditional mechanics sourcebooks. The building of physics on the variational and symmetry principles gives several advantages in their hands. Variational methods are suitable to describe any field, with different charges and forces. Therefore, this treatment - over the classical mechanics - can incorporate the Maxwell equations for electric and magnetic fields, the Schrödinger equations of wave mechanics and even the Klein-Gordon wave equations for infinite dimensional systems. Thus classical mechanics, electromagnetic (and any) fields as well as quantum mechanics can be treated by the same tools, as a unit, alike nature is also not divided into parts by different physical phenomena.

The authors emphasize the central role of symmetry in their treatment: "Symmetry and mechanics have been close partners since the time of the founding masters: Newton, Euler, Lagrange, Laplace, Poisson, Jacobi, Hamilton, Kelvin, Routh, Riemann, Noether, Poincaré, Einstein, Schrödinger, Cartan, Dirac, and to this day, symmetry has continued to play a strong role, especially with the modern work of Kolmogorov, Arnold, Moser, Kirillov, Kostant, Smale, Souriau, Guillemin, Sternberg and many others. This book is about these developments, with an emphasis on concrete applications that we hope will make it accessible to a wide variety of readers, especially senior undergraduate and graduate students in science and engineering." Really, if one has asked a physics student where he/she has been made acquainted with the application of symmetry in physics, they would have rarely mentioned just (classical) mechanics. Nevertheless "symmetry was already widely used in mechanics by the founders of the subject, and has been developed considerably in recent times in such diverse phenomena as reduction, stability, bifurcation and solution symmetry breaking relative to a given system symmetry group, methods of finding explicit solutions for integrable systems, and a deeper understanding of special systems, such as the Kowalevski top."

The authors had several prior publications in the topic. J. E. Marsden is professor of control and dynamical systems at Caltech. He is also one of the editors of the Text in Applied Mathematics series (since 1982; for the readers of the VisMath: together with M. Golubitsky, W. Jäger and L. Sirovich). T. R. Ratiu is professor of mathematics at the UC Santa Cruz and the Swiss Federal Institute of Technology, in Lausanne. Since the size of the printed book was limited, the printed text and its visualization is extended on the Internet (free access at http://www.cds.caltech.edu/~marsden/books/). Updates, solution manual for instructors, and further information, including errata can be downloaded too.

The book starts with an introduction to the Lagrangian and Hamiltonian formalisms. (Here they emphasize the role of mechanics influenced on the development of mathematics.) Among the demands for mathematical tools, the role of symmetry is underlined "from fundamental formulations of basic principles to concrete applications, such as stability criteria for rotating structures. The theme of this book is to emphasize the role of symmetry in various aspects of mechanics." (p. 1) Then they continue with the theory in linear spaces, in which case the symplectic form becomes a skew-symmetric bilinear form that can be studied by linear-algebraic methods. Later they generalize the symplectic structures to manifolds. In this course they base Lagrangian mechanics primarily on variational principles rather than on symplectic or Poisson structures. They show also its equivalence to the Hamiltonian under appropriate hypotheses. First, as an example, they show how is Newton’s second law equivalent to Hamilton’s canonical equations, which is a first order system in a phase-space (p. 63), before they introduce the symplectic and Poisson structures with the help of the Hamiltonian vector field and the symplectic matrix. So they reach to the definitions of the canonical transformation, the symplectic transformation and Poisson transformation, as well as to the symplectic group (pp.70-72). Thus, the student is introduced step by step in the canonical and group theoretical (algebraic) description of mechanics interpreted in vector fields.

Later they need no more, than to develop the program. Following the well-based (by definitions and tools) mathematical basis, the full mechanics can be easily explained. So, it is not surprising, that the next step, the introduction of the description of the conserved quantities by the Poisson and Schrödinger brackets is only an example for the learned method (p. 117), alike the linear and angular momenta and the Schrödinger equations (pp. 118-119).

Later on a special chapter is devoted to the "basic facts about Lie groups", based also on the linear and angular momenta, as conserved quantities associated with the groups of translations and rotations in space (p. 265). They set group SO(3) to be the 3D Lie group, which serves as the configuration space. This group plays also a (double) role of a symmetry group which leads to conservation of the angular momentum of a rigid body. The role of the Lie group and the linear and angular momenta are first shown on the examples of the rigid body, the heavy top, incompressible fluids, compressible fluids, megnetohydrodynamics, the Maxwell-Vlasov equations and the Maxwell equations.

A next chapter shows, how one can obtain "conserved quantities for Lagrangian and Hamiltonian systems with symmetries." The book uses the help of the concept of "momentum mapping", which is geometric generalization of the classical linear and angular momenta. "This concept is more than a mathematical reformulation of a concept that simply describes the well-known Noether theorem. Rather, it is a rich concept that is ubiquitous in the modern developments of geometric mechanics." (p. 365)

The book devotes a special chapter to two processes: one called as Lie-Poisson reduction, as well as to the other side of the story – where the basic objects that are reduced are not Poisson brackets but rather are variational principles – which process is called as Euler-Poincaré reduction.

The last chapters of the book present classical mechanical examples for the application of the approaches, methods and tools used in the systematic chapters.

The volume contains a very detailed (32 page) bibliography and a 30 pages Index. Unfortunately, the Index is useless, since the referred page numbers in the majority of cases do not refer to the real pages, where the given subject can be found. This was later corrected and the errata can be downloaded from the first author’s homepage.
 
Reviewed by György Darvas