Paulette Libermann's book

Paulette Libermann and Charles-Michel Marle wrote the four volume work Géométrie symplectique, bases théoriques de la mécanique (1986, 1986, 1987). It was translated from the French by Bertram Eugene Schwarzbach and the English translation published as the single volume Symplectic geometry and analytical mechanics, by Paulette Libermann and Charles-Michel Marle (Reidel Publishing Company, 1987). We give various extracts describing the work.

Symplectic geometry and analytical mechanics, by Paulette Libermann and Charles-Michel Marle.

Dedication.

... à la memoire d'Élie Cartan et de Charles Ehresmann en hommage à Andre Lichnerowicz.
From the Series Editor's Preface.
Approach your problems from the right end and begin with the answers. Then one day, perhaps you will find the final question.
The Hermit Clad in Crane Feathers in R van Gulik's The Chinese Maze Murders.

It isn't that they can't see the solution. It is that they can't see the problem.
G K Chesterton. The Scandal of Father Brown 'The point of a Pin'.
Growing specialisation and diversification have brought a host of monographs and textbooks on increasingly specialised topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related.

Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowski lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This program, Mathematics and Its Applications, is devoted to new emerging (sub)disciplines and to such (new) interrelations as 'exempla gratia':
- a central concept which plays an important role in several different mathematical and/or scientific specialised areas;

- new applications of the results and ideas from one area of scientific endeavour into another;

- influences which the results, problems and concepts of one field of enquiry have and have had on the development of another.
The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practitioners in diversified fields.

The love affair, or marriage, between geometry and physics, here mainly in the guise of analytical mechanics, is, as is well known, an old one. There seems to have been a period of cooling off. However, after a trial separation period the relations between the two are more vigorous and deeper than ever before.

The branch of geometry chiefly concerned is (generalised) symplectic geometry. Perhaps surprisingly at first sight, given its age (though not really so if one pauses to reflect on the demands made by new developments on the application side), this continues to be a very active field with many recent new results and concepts. For instance, the notion of a Poisson manifold as a naturally appearing generalisation of a symplectic one is quite possibly a more fundamental and more supple tool.

This book specifically aims to present these new developments integrated with the more well-established frameworks of symplectic geometry and, as such, is a timely and, I am convinced, most valuable addition to the existing selection of books on symplectic geometry.
The unreasonable effectiveness of mathematics in science ...
Eugene Wigner

Well, if you know of a better 'ole, go to it.
Bruce Bairnsfather

What is now proved was once only imagined.
William Blake

As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company they drew from each other fresh vitality and thence forward marched on at a rapid pace towards perfection.
Joseph Louis Lagrange
From the Authors' Preface.

During the last two centuries, analytical mechanics have occupied a prominent place among scientists' interests. The work in this field by such mathematicians as Euler, Lagrange, Laplace, Hamilton, Jacobi, Poisson, Liouville, Poincare, Carathéodory, Birkhoff, Lie and E Cartan has played a major role in the development of several important branches of mathematics: differential geometry, the calculus of variations, the theory of Lie groups and Lie algebras, and the theory of ordinary and partial differential equations. During the last thirty years, the study of the geometric structures which form the basis of mechanics (symplectic, Poisson and contact structures) has enjoyed renewed vigour. The introduction of modern methods of differential geometry is one of the reasons for this renewal; it has permitted a formulation of global problems and furnished tools with which to solve them.

Even though there are already a number of books that treat this subject, the authors believe that it is of value to provide readers with an approach to these methods and to permit them to familiarise themselves with certain recent developments which are not mentioned in the other textbooks in this field, and to acquire the information necessary in order to pursue current research. They have also expounded and employed the methods of exterior algebra which were introduced by E Cartan.

This work, which is in large part based on lectures given by the authors at the Universities of Paris VI and VII, incorporates numerous points recalled to facilitate study. It was written for students at the end of their 'Second Cycle' programme, or at the beginning of their 'Third Cycle' - these are the French designations that correspond, approximately, to American Masters' and Doctoral programs, respectively. It is mainly directed at readers interested in mathematics, but it may be of interest too for physicists, engineers, and for anyone who may be interested in differential geometry and the foundations of mechanics.

This work is composed of five chapters and seven appendices. Chapters I, II and V, as well as appendix 2, were written by the first author (Paulette Libermann), while chapters III and IV, as well as the remaining appendices, were written by the second author (Charles-Michel Marle).

The first chapter is composed of three parts. The first part deals with symplectic vector spaces (rank, orthogonality, the linear symplectic group, reduction, the contravariant point of view). These notions are taken up again in the second part which treats symplectic vector bundles. Finally, the third part discusses, in an abbreviated fashion, Lepage's theorems concerning the decomposition and the divisibility of forms on a vector space or a differentiable manifold.

The second chapter concerns semibasic forms, especially the Liouville form on the cotangent bundle. It includes a discussion of the Legendre transformation and its applications to mechanics. Though it adopts a different point of view, this chapter has been suggested by J Klein's formulation of mechanics in terms of the tangent bundle, which has been presented in C Godbillon's book.

The third chapter deals with symplectic structures on a manifold and with their generalisations: presymplectic structures and, especially, Poisson structures. It has been known for many years that symplectic manifolds offer a suitable framework for the Hamiltonian formulation of classical mechanics. Poisson manifolds, which were discussed by S Lie as early as 1890, and, implicitly, recognised by E Cartan, appear naturally, for example in the study of certain foliations of symplectic manifolds. This chapter also includes a thorough discussion of the reduction of a symplectic manifold, and gives a complete account of Weinstein's generalisations of Darboux's classical theorem. Then it proves Liouville's theorem regarding completely integrable systems, as well as the theorem of Arnold and Avez regarding the existence of action-angle coordinates.

The fourth chapter deals with the action of a Lie group on a symplectic manifold. The important notion of the momentum map, which is due to J-M Souriau and S Smale is defined and its properties are studied. The special case in which the symplectic manifold is the cotangent bundle of the Lie group is of particular interest, and leads quite naturally to the definition of the Lie-Poisson structure on the dual space of the Lie algebra of a group. This structure, recognised by S Lie around 1890, has been recently rediscovered by A Kirillov, B Kostant and J-M Souriau. The results concerning the reduction to a symplectic manifold upon which a Lie group acts are applied to the definition of the stationary motions of a Hamiltonian system which possesses a Lie group G of symmetry and to the study of their relative stability modulo G. The chapter concludes with an application of all the notions that were previously introduced to the classical example of the motion of a rigid body about a fixed point.

The fifth chapter deals with contact structures and, more generally, with Pfaffian equations of arbitrary class. We prove the theorems of Darboux relative to Pfaflian forms and equations of constant class. A large part of the chapter, which was suggested by the work of A Lichnerowicz and V Arnold, concerns the symplectification of a compact manifold. These considerations lead to the notion of a Liouville structure on a principal bundle, whose structure group is the multiplicative group $\mathbb{R}^{*}$ , equipped with a Pfaflian form without zeros that is homogeneous of degree 1. Contact geometry has applications to time-dependent Hamiltonian systems, and to the homogeneous Hamiltonian systems encountered in mechanics. Lastly, the Legendre transformation is studied from the point of view of contact structures.

Appendices 1 (basic notions in differential geometry), 4 (integral invariants) and 5 (Lie groups and Lie algebras) deal with basic notions which are used in the book. Readers already familiar with these subjects may skip them or merely refer to them to become acquainted with our notations. Appendix 3 (distributions, Pfaflian systems and foliations, includes a review of well-known notions, and offers a generalisation of Frobenius' theorem due to P Stefan and H Sussmann, which is applicable to distributions with nonconstant rank. This is useful in the study of Poisson manifolds. The remaining appendices develop subjects, already encountered in the body of the book, which are interesting in their own right: the application of the infinitesimal jets of C Ehresmann to mechanics (appendix 5), Lagrange-Grassmann manifolds (appendix 6), the use of Morse families in the definition of Lagrangian submanifolds of a cotangent bundle, first-order partial differential equations, Jacobi's theorem, and the Hamilton-Jacobi equation (appendix 7).

Several important aspects of the mathematical structures of analytical mechanics could not be included in this book, or had to be treated all too briefly. Among them we must mention the Lagrangian formulation of mechanics and the calculus of variations (touched on in chapter II), holonomic and nonholonomic constraints in mechanical systems, the formulation of canonical transformations and of the Hamilton-Jacobi method in the framework of the canonical manifolds of A Lichnerowicz, the symplectic relations defined by W M Tulczyjew (barely mentioned in chapter III), the study of the properties of the Arnold-Leray-Maslov index (whose construction is sketched in appendix 6). In these case, precise references have been supplied to compensate for the gaps.
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During the writing of this work Andre Lichnerowicz and Georges Reeb lavished advice and encouragement upon us, and we are pleased to express our gratitude.
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Review by: I Vaisman.
Mathematical Reviews MR0882548 (88c:58016).

The present work is an advanced textbook which gives a systematic exposition of the theory of symplectic, Poisson and contact manifolds, and their applications in Hamiltonian mechanics. The basic text consists of five chapters whose content is as follows. Chapter I, "Symplectic vector spaces and symplectic vector bundles", treats vector spaces and bundles endowed with a nondegenerate skew-symmetric bilinear form including such items as: symplectic forms, the symplectic group, subspaces and Lagrangian subspaces, reduction, adapted complex structures, and, also, the Libermann-Lepage decomposition of exterior forms and Darboux's theorem. Chapter II, "Semibasic and vertical differential forms in mechanics", treats the symplectic structure of cotangent bundles, the theory of the Legendre transformation, Lagrange and Hamilton equations of mechanics, calculus of variations, time-dependent mechanics. Chapter III, "Symplectic manifolds and Poisson manifolds", defines and studies the basic notions such as symplectic forms and manifolds and their submanifolds, Hamiltonian vector fields and Poisson brackets, Poisson manifolds and morphisms (notice the first appearance of this recent subject in a textbook), the reduction theory, the Darboux-Weinstein theorem, completely integrable Hamiltonian systems. Chapter IV, "Action of a Lie group on a symplectic manifold", treats Hamiltonian group actions, momentum maps, reductions, and develops applications in mechanics, particularly to the motion of a rigid body. Chapter V, "Contact manifolds", gives the geometric framework of time-dependent mechanics, and discusses contact manifolds and their symplectifications and related structures, Liouville structures, homogeneous Hamiltonian systems, time-dependent systems, etc. Several appendices provide as a supplement basic differential geometry (jets, distributions and foliations, Lie groups and algebras, etc.) and further developments of symplectic geometry. The book, which was written by two well-known specialists with important contributions in the field (also reflected in this book) is very well written, updated, and it supplies a valuable contribution to the geometric-mechanical literature.
Review by: Tudor Ratiu.
American Scientist 78 (3) (1990), 282-283.

This book presents the basic notions of geometric mechanics in a coherent and self-contained manner specifically for graduate students in mathematics. The book serves also as a very convenient reference work. It has an excellent index and is organised in such a way that locating specific theorems and corresponding concepts is easy-quite an achievement in a text of this size.

A rough outline of the literature on this subject will explain to some extent the authors' choice of material and point of view. After the early books of Mackey (1963) and Sternberg (1963), a first generation of books appeared between 1969 and 1973 which collectively set the tone for all subsequent research in symplectic geometry, Hamiltonian dynamics, and their applications. These books fell into roughly two categories: those emphasising the geometric aspect and those stressing the dynamic. This unfortunate division was dictated mainly by considerations of space; the material necessary to cover both aspects was prohibitively extensive. Ten years later, a second generation of works appeared, some trying to bridge the gap, others simply confining themselves to treating special aspects in detail.

Major developments in symplectic and Poisson geometry as well as in dynamics warrant a third generation of books to bring the researcher up to date. By now, however, this has become an utterly impossible task. There are only two ways out: either present a very precise synopsis of the subject, more or less in encyclopaedia form, or focus on very specific topics. So it comes as no surprise that the third generation is opened by three volumes: Arnold, Kozlov, and Neishtadt's Dynamical Systems (1988), the third volume of Springer's Encyclopaedia of Mathematical Sciences, which surveys the whole field; Fomenko and Trofimov's Integrable Systems on Lie Algebras and Symmetric Spaces (1988), which deals in depth with a very important class of integrable systems; and this volume, in which the authors decide on very specific geometric goals and honestly state what they will omit.

The book contains eight chapters together with seven appendixes whose sole purpose is to keep the cross-referencing of basic material or work not available in book form to a minimum. The appendixes also help the reader untangle almost immediately the various notations and symbols used by the authors in tensor analysis on manifolds and fibre bundles. Chapter 1 covers basic symplectic algebra, chapter 2, Hamiltonian mechanics on the cotangent bundle and Lagrangian mechanics on the tangent bundle. Chapter 3 treats symplectic and Poisson manifolds abstractly; this is the first time the basic properties of Poisson manifolds as described by Lichnerowicz and Weinstein have appeared in book form. This chapter also discusses the abstract theory of completely integrable systems and action-angle variables and contains the only slip I noted in the references: the paper by Jost improving on the Arnold-Avez construction of the action-angle variables is not cited even though the result is presented.

Chapter 4 deals with the theory of symplectic reduction. Unfortunately, it does not elaborate on reduction of cotangent bundles, magnetic terms and their relationship to connections, reductions leading to semidirect products, or the theory of reduction in the Poisson case. But it does present the basic theory of mutually orthogonal or dual pairs, which is central to the geometric ideas surrounding reduction. Chapter 5 deals with contact manifolds and time-dependent systems, but only in the context of a cotangent bundle times the real axis. It would have been quite interesting to have a discussion of the analogue of Hamiltonian systems on contact and Jacobi manifolds in general.

The positive aspects of the book are the clarity of the exposition, the excellent cross-referencing, the comments on the literature, and the accessibility of results. The main weakness is the scarcity of examples, which gives the book an undeserved aura of very abstract pure mathematics. Had the authors included more examples and actively discussed them rather than relegating them to the remarks on the literature, the book would have appealed to a wider audience, especially theoretical physicists and engineers. I also found the algebraic aspect overemphasised, given the authors' slated goals. The inclusion of some of the material in chapter 1 without a follow-up on more serious aspects of Lagrangian manifolds or the theory of the Maslov index, for example, is hard to justify. Given the impossibility of doing justice to even a reasonable fraction of the recent developments, however, this is a minor inconvenience. Within its aims, the book is a valuable addition to the literature on geometric mechanics.
Review by: William M Boothby.
Bull. Amer. Math. Soc. 20 (1989), 89-94.

This is the most recent of several books appearing over the last fifteen years or so which well may help to restore analytical mechanics to the important place it once occupied in the training of mathematicians. Courses in this subject, then known as "rational mechanics", were standard fare in many mathematics departments until about 50 years ago, at which time they virtually disappeared and the subject went underground for a generation of mathematicians - at least as far as graduate study in the United States was concerned. Of course, it did not disappear as a topic of research, and in fact this spate of excellent books has surely resulted in large part from the profound and exciting research which has been done in analytical mechanics or has been suggested by it over this same period, and in some part too from the tremendous advances in related manifold theory and differential geometry, which have revolutionised the approach to mechanics and made much of this work possible. Happily there now exist several extremely well written and carefully designed texts from which a graduate student or working mathematician can learn this important subject, the well-spring of so much mathematics, and can even come up to the very frontiers of research. Of these the books of Arnold and Abraham and Marsden may be the most frequently used and best known (in English). To this I would add one of the earliest of this breed, Godbillon, which like the other two is extremely clear and well written but which because of its more limited scope has the advantage (especially for the beginner who wants a quick start) of being very much shorter. The present book of Libermann and Marie, which is based on courses given to graduate students at the University of Paris VI and VII, is very similar in level and "prerequisites" to these three. In style and treatment it is most like Godbillon, but in scope and content much closer to Arnold and Abraham and Marsden, with both of which it has a great deal of overlap. But there are also important differences. Its emphasis is very much on the mathematical underpinnings of mechanics and, with one exception, there is almost no discussion of physical examples as there is in both Abraham and Marsden and, especially, Arnold. Thus it has its own style, approach, and priorities, which give it an emphasis different from the others. To me this makes it a useful complement to other books with the same objective - making analytical mechanics and its related research available to the mathematically trained - and I find it a welcome addition to the collection of books with this goal.

In its modern reincarnation, more than ever before, mastery of this subject requires an interest in or willingness to learn modern differential geometry and manifold theory. To one who already has some knowledge of these subjects, analytical mechanics serves as a fascinating and concrete example of many concepts which may have been learned in a rather abstract setting, and as an important lesson in the historical background of manifold theory (which owes so much to Poincaré's work in mechanics). To one who is not already skilled in modern differential geometry, these books present well-arranged and carefully thought out introductions to the subject in a highly motivated context.

Really excellent reviews of [V I Arnold, Mathematical methods of classical mechanics; and R Abraham and J E Marsden, Foundations of mechanics] by reviewers who are themselves important contributors to the subject have appeared in this journal. They contain a great deal of historical background and, in particular, Sternberg's review [of R Abraham and J E Marsden] contains an interesting account of the main themes of mathematical research, which I mentioned above as having so revitalised this classical subject. These reviews should certainly be read by anyone who wants to get a feel for the available texts up to 1980 before launching into a study of analytical mechanics. It would be superfluous and even presumptuous of me to attempt to cover the same material, so I will limit myself to just enough background to explain some of the emphasis of the book being reviewed and to try to contrast its contents with the others.
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we see that any book or course which has the purpose of making analytical mechanics accessible to the general mathematical reader must first of all present the differential geometric foundations - vector fields on manifolds, vector bundles over manifolds, the geometry of exterior differential forms, symplectic geometry, etc. - as carefully and understandably as possible. This is done across an interesting spectrum of approaches in each of the books mentioned. At one end is Arnold, the most concrete and example-oriented: the mathematical concepts are introduced only as they are needed to develop the examples from mechanics. At the other end Libermann and Marie have very few mechanical examples, and then only after the mathematics has been very thoroughly developed. Abraham and Marsden is in the intermediate position. Which approach one prefers is very much a matter of personal taste and objectives, the way one prefers to learn a new subject, and where one is starting from. I find it very pleasant to have the choice of (at least !) four well-written books to skip around in.

With this introduction we turn to a brief description of Libermann-Marle's book itself. It contains five chapters to which have been added seven appendices and a thirty page bibliography. Of the three chapters written by Libermann the first contains a very complete presentation of symplectic geometry, vector bundles, and exterior differential forms. As might be expected of a distinguished practitioner of exterior calculus, there is much more on this subject than can be found in any of the other books on mechanics, e.g. the LePage decomposition theorems in the first chapter and then a complete chapter, Chapter V, on contact forms and structures. Chapter II puts all of this in the setting needed for mechanics, discussing variational principles, deriving Lagrange's and Hamilton's equations and explaining the important concept of a Legendre transformation, which forms the bridge between the Lagrange and the Hamilton approach, i.e. between $T(M)$ and $T^{*}(M)$ . None of this basic material is easy, even for someone with a general knowledge of manifold theory, but the exposition of the theory here is clear and careful. There is, to be sure, extensive overlap with the presentation in other texts such as those mentioned above, but there is also a good deal of interesting material not found in them. In Chapters III and IV, written by Marie, we find a detailed study of symplectic manifolds, Poisson manifolds, the Darboux-Weinstein theorems and a wealth of related material. Chapter IV is completely devoted to the action of a Lie group on a symplectic manifold. It covers, in particular, the momentum map of Souriau and Smale and leads into the work of Kirillov, Kostant and Souriau mentioned earlier and ends with an analysis of the motions of a rigid body in several specific cases - the Euler-Poinsot, Euler-Lagrange and Kowalevska problems. Both of these chapters have problem sets.

The seven appendices contain some background material on basic differential geometry, distributions and pfaffian systems, Frobenius Theorem (as generalised by Stefan and Sussmann), foliations, and Lie groups, for example. But they also contain several important supplements to the text: jets, Lagrange-Grassmann manifold, Morse families and Lagrangian submanifolds, and an appendix on integral invariants. There are a number of interesting comments and historical notes throughout the text, including some discussion of topics which have been omitted accompanied by references to the relevant literature. A study of this book, or even selected parts of it, will surely bring the reader to the point at which the research articles listed in the extensive bibliography are quite accessible, which is indeed one of its aims. Finally, it should be mentioned that although originally written in French, the translation by B E Schwarzbach is very good and the exposition throughout is thorough and careful. This book is a welcome addition to the literature. It will surely prove useful as a reference, a place to learn the subject, or as an adjunct to the earlier books mentioned above.
Review by: N M J Woodhouse.
Bull. London Math. Soc. 20 (4) (1988), 377-379.

As Libermann and Marle remark in their preface, the introduction of modern geometric methods has given classical mechanics a renewed vigour during the last thirty years. It has brought into sharp focus the structures underlying the great eighteenth and nineteenth century advances in analytical dynamics; and it has opened the way for new applications in mathematical physics, in group theory, and in the analysis of partial differential equations.

Libermann and Marle are not the first to reveal the magic of Lagrange and Hamilton stripped of the clutter of 'generalised coordinates' and 'infinitesimal displacements'; and their book is certainly not the most comprehensive. But it is, nevertheless, very welcome. They do not attempt to cover the whole range of analytical dynamics (their principal topics are symplectic linear algebra; Lagrange's and Hamilton's equations; presymplectic, symplectic, Poisson, and contact structures; reduction; and the momentum map); but they develop the theory with great care and thoroughness. Darboux's theorem, for example, they prove in three different ways: first by using results about the decomposition of differential forms; second as a corollary of a more general statement about the local structure of Poisson manifolds; and third by Weinstein's method.

In spite of the wealth of detail (much of which is either new or not easily accessible elsewhere) the book is readable, extremely clear, and precise without being fussy. It is refreshingly free from idiosyncratic notation - although 'orth $W$ ' for the (symplectic) orthogonal complement of a subspace $W$ looks distractingly odd in the English text; and enough use is made of coordinates for it to be easy to make contact with the classical literature. It will therefore be invaluable as a reference work as well as a course text. The first five appendices, which summarise in just over 100 pages some basic ideas in differential geometry and Lie group theory, are a useful contribution in their own right.

The authors have clearly achieved their main aim: to give a systematic, accurate and largely self-contained account of the geometric foundations of classical mechanics for graduate students in mathematics. In doing so, they have written an excellent and lasting book.

They are less successful, however, when its comes to applications. It is clear that their meticulous account of the basic theory left them with little space for examples. They have compensated to some extent for the omission of applications within mathematics by including appendices on such topics as Morse families and the Lagrangian Grassmannian. But specific examples of mechanical systems are thinly scattered in the text: the various classical 'completely integrable' cases of rigid body motion are the only systems dealt with in any detail. Even these are seen as being of interest more as illustrations of the theory than as physical systems and are treated somewhat pedantically (for example, the definition of the total mass of rigid body requires a reference to Halmos's Measure theory). This is a pity: as Arnold has shown, the interaction between geometry and mechanics is not a one-way process. The new geometric methods also yield valuable physical insights.

Like Lagrange's Mécanique analytique, Libermann and Marle's book contains not a single diagram - a surprising omission considering that 'geometry' and 'mechanics' both appear in the title.

Last Updated December 2021