# Presentation of Jean-Marie Souriau’s book
“Structure des systèmes dynamiques”

Géry de Saxcé
Emeritus Professor
Univ. Lille, CNRS, Centrale Lille, UMR 9013 – LaMcube –
Laboratoire de mécanique multiphysique multiéchelle, F-59000, Lille, France

Charles-Michel Marle
Honorary Professor
retired from the former Université Pierre et Marie Curie
today Sorbonne Université, Paris, France

###### Abstract

Jean-Marie Souriau’s book “Structure des systèmes dynamiques”, published in
1970, republished recently by Gabay, translated in English and published under
the title “Structure of Dynamical Systems, a Symplectic View of Physics”, is a
work with an exceptional wealth which, fifty years after its publication, is
still topical. In this paper, we give a rather detailled description of its
content and we intend to highlight the ideas that to us, are the most creative
and promising.

Jean-Marie Souriau’s book “Structure des systèmes dynamiques” [1] was
published in 1970 in a book collection for students in the first year of
master’s degree in mathematics. It is directed in fact to mathematicians,
beginners or experienced, wishing to know the applications of mathematics to
physical sciences, and to physicists concerned with knowing certain
mathematical tools useful for their researches. The author was very aware of
this since, in his Introduction, he gives reading recommendations adapted to
both reader categories. It is a work with an exceptional wealth which, fifty
years after its publication, is still topical. In the first part, we shall be
describing its content. In the second one, we shall discuss the most original
aspects. As conclusion, we shall indicate why this book seem to us still today
a valuable source for the students and researchers in mathematics, mechanics
and physics.

##  1 Book content

###  1.1 A quick survey.

This book comprises a rather large introduction (20 pages) and five chapters.
The first two chapters, entitled _Differential geometry_ and _Symplectic
geometry_ , are purely mathematical. The third one, entitled _mechanics_ ,
begins with a classical presentation of the mechanics of material point
systems. Very soon, the author introduces the concept of _manifold of motions_
of a system. Then he applies the methods of symplectic geometry presented in
the previous chapter both to classical and relativistic mechanical systems. An
important paragraph deals with isolated systems admitting a symmetry group,
acting transitively on the manifold of motions, that the author considers as
models of _elementary particles_. Next the author studies the dynamics of
systems of elementary particles, taking into account their interactions. The
fourth chapter, entitled _statistical mechanics_ , contains two sections. The
first one, essentially mathematical, presents measure and integration
theories, together with notions of probability theory. The author defines the
_statistical states_ of a dynamical system, the _entropy_ of a statistical
state and proposes a generalisation of the notion of _Gibbs state_. In the
second paragraph, he uses these notions to treat certain systems seen in
physics: classical and relativistics ideal gas, systems of null mass
particles, to name a few. He clarifies the interpretation of the parameters
which the Gibbs state depends on in terms of thermodynamical quantities, and
proposes an interesting generalisation of the concept of thermodynamical
equilibrium. The fifth and last chapter, entitled _Geometric quantization_ ,
presents a construction which allows to associate to a symplectic manifold
satisfying certain conditions another manifold, of larger dimension, called by
the author a _quantum manifold_. This construction is due to the author. A
slightly different but equivalent construction was proposed independently by
the American mathematician B. Kostant [2]. In the second and last section of
the fifth chapter, the author applies this construction to the quantization of
physical systems.

The book is illustrated by many figures which mostly are very meaningful
schematic representations of the geometric constructions used by the author.
The references to other books or papers, fairly limited in number, are not
gathered in a bibliographic list, but indicated in footnotes. An index, very
detailled and easy to use, and the list of main notations completes the book.

###  1.2 Detailed presentation

#### The introduction.

The author evocates the book by Joseph-Louis Lagrange (1736–1813) _Mécanique
analytique_ [3], written at the end of the 18th century. This famous work is
at the origin of the _mécanique analytique classique_ which was, during the
19th century and the first half of the 20th century, an essential part of the
scientific learning in the French universities and high schools. According to
the author, it is an unfinished work in which some chapters are only sketched.
For him, the form used to present this theory and the concepts it uses
(instead of concepts, the author writes the _categories_ in the
epistemological, Aristotelian or Kantian sense) were fixed by Lagrange’s
successors such as Siméon Denis Poisson (1781–1840), William Rowan Hamilton
(1805–1865) and Carl Gustav Jakob Jacobi (1804–1851). The author considers
that the form thus given to analytical mechanics, although it gives to this
theory a formal mathematical perfection, has lost an important part of
Lagrange’s thought. The great discoveries of the first quarter of the 20th
century (special and general relativity, quantum theory) taught us that the
words _time_ , _space_ and _matter_ not necessarily have the obvious meaning
ascribed thereto. For the author, classical analytical mechanics, which
remains an essential ingredient of current physical theories, is not outdated,
although certain concepts it uses are so because they do not have the required
covariance, in other words because they are in contradiction with Galilean
relativity. He wishes to show in his book that a better consideration of
Lagrange’s thought allows a formulation of this theory compatible with the
most recent discoveries in physical sciences.

Using the very simple example of the motion of a material point, the author
explains which concepts he is going to use: evolution space, space of motions
of a system, Lagrange’s form. The concept of space of motions of a dynamical
system seems to us the most important : it is the set of all possible motions
of the considered system. The author discusses in depth its usefulness, often
underestimated by scholars mostly interested in the study of one particular
motion of the considered system. He presents also all the mathematical tools
that he will use: differential forms, Lie groups, symplectic forms, etc. Hence
he reviews fairly accurately every chapter of his book and completes his
introduction with advises to the readers.

#### The first chapter, Differential Geometry

This chapter presents in less than 70 pages numerous delicate notions:
differential manifolds, tangent and cotangent fiber bundles, submanifolds,
covering spaces, vector fields and differential equations, Lie bracket of two
vector fields, exterior derivative, foliations, Lie groups, calculus of
variation. The author presents the concept of differential manifold in a
rather original way which does not use the previous presentation of
topological manifolds, undoubtedly to make this notion easily accessible to
beginning students. Differential manifolds are not assumed to be Hausdorff.
The _space of motions_ of certain mechanical systems encountered in Chapter
III are indeed non-Hausdorff manifolds. The author’s language sometimes
slightly differs from that generally used : for example, he calls an
_embedding_ what most of geometers call an _injective immersion_. However,
readers can easily avoid misunderstandings, since the author scrupulously
defines all the terms he uses.

The paragraph devoted to Lie groups contains a detailed presentation of the
actions of a Lie group on a differential manifold and of its adjoint
representation. The author will define the coadjoint representation in the
next chapter, with the study of the moment map of the action of a dynamical
group on a presymplectic or symplectic manifold. The main classical Lie groups
(linear, orthogonal, unitary, symplectic) are described in a very original
manner. The section about calculus of variations presents, besides the
classical Euler-Lagrange equations, the extremality criterion using Cartan’s
form, often called _Euler-Cartan theorem_ , that establishes a link between
calculus of variation and symplectic geometry, together with Noether’s
theorem.

#### Chapter II, Symplectic Geometry.

It is also essentially mathematical. Clearly shorter than the previous one (46
pages), it begins with the study of a finite dimensional vector space $E$
equipped with a bilinear skew-symmetric form $\sigma$. The author defines the
concepts of _orthogonality_ with respect to $\sigma$, of _isotropic_ _co-
isotropic_ and _self-othogonal_ vector subspaces, also called by other authors
_Lagrangian vector subspaces_ when $\sigma$ is nondegenerate. The author
proves that the rank of $\sigma$ is always even and that given a coisotropic
vector subspace, one can always build a basis of the kernel of $\sigma$, then
complete it to obtain a basis of $E$, called _canonical_ , in which the
expression of $\sigma$ is very simple. When $\sigma$ is non degenerate, it is
called a _symplectic form_ , the dimension of $E$ is even and the couple
$(E,\sigma)$ is called a _symplectic vector space_. Hence the author defines
the _symplectic group_ of $(E,\sigma)$ and studies its natural action on $E$,
its features, together with thoses of its Lie algebra. He shows in particular
that $E$ has a complex vector space structure adapted, in a precise
mathematical sense, to the symplectic form $\sigma$, and that it can be
endowed with a Hermitian form of which $\sigma$ is the imaginary part.

The author defines next symplectic and presymplectic manifolds and studies
their properties. He shows that under certain conditions, the quotient of a
presymplectic manifold by the kernel of its presymplectic form is a symplectic
manifold, a result that he will use in the following chapter to define the
_space of motions_ of a dynamical system. He shows that to each differentiable
function defined on a symplectic manifold, there is an associated vector field
that he calls its _symplectic gradient_. The flow of this vector field leaves
unchanged the symplectic form. He proves that the set of the differentiable
functions defined on a symplectic manifold is endowed with a binary operation,
the _Poisson bracket_ , that makes it a Lie algebra of finite dimension. He
defines and studies some remarkable submanifolds of a symplectic manifold
(isotropic, co-isotropic and self-orthogonal submanifolds), studies their
properties and gives several examples thereof. He proves _Darboux theorem_ ,
whereby every point of a symplectic manifold is an element of the domain of a
chart, called _canonical_ , in which the symplectic form is expressed in a
simple manner. Next he defines and studies the _symplectomorphisms_ , also
called _canonical transformations_ , and their generalisations, the _canonical
similarities_ , together with the _infinitesimal canonical transformations_
(called by other authors _locally Hamiltonian vector fields_).

The author calls _dynamical group_ of a symplectic or presymplectic manifold a
Lie group acting on it by canonical transformations. He calls _moment_ _map_
of the dynamical group $G$ of a presymplectic or symplectic manifold $M$ a
smooth map $\Psi$ from $M$ into the vector space ${\mathcal{G}}^{*}$, dual of
the Lie algebra $\mathcal{G}$ of $G$, such that for every $Z\in{\mathcal{G}}$,
the infinitesimal generator of the action on $M$ of the one-parameter subgroup
generated by $Z$ is the symplectic gradient of the function linking, at every
$x\in M$, the real $\bigl{\langle}\Psi(x),Z\bigr{\rangle}$. In the terminology
used by most of geometers, this infinitesimal generator is called the
_Hamiltonian vector field_ whose Hamiltonian is the function
$x\mapsto\bigl{\langle}\Psi(x),Z\bigr{\rangle}$.

The author gives several examples of dynamical groups, indicates sufficient
conditions for the existence of a moment map and studies its properties. This
leads him to propose a generalisation of Noether’s theorem encountered in the
part of the previous chapter concerning the calculus of variations. After a
quick presentation of the cohomology of Lie groups and algebras, the author
shows that to any moment map of the action of a dynamical group $G$ on a
connected presymplectic or symplectic manifold, there always exist an
associated cocycle $\theta$ of $G$ valued in the dual ${\mathcal{G}}^{*}$ of
its Lie algebra. The differential of $\theta$ at the neutral element, which is
the cocycle of the Lie algebra ${\mathcal{G}}$ associated to $\theta$, is a
skew-symmetric bilinear form on ${\mathcal{G}}$. The author calls _symplectic
cocycle_ a cocycle of $G$ valued in ${\mathcal{G}}^{*}$ satisfying this
property. He proves that the addition to the moment map of a constant element
of ${\mathcal{G}}^{*}$ modifies the cocycle $\theta$ by addition of a
coboundary, therefore does not change its cohomology class, which depends only
on the $G$-action, not on the choice of the moment map. Moreover the cocycle
$\theta$ allows to define an affine action of $G$ on the dual
${\mathcal{G}}^{*}$ of its Lie algebra whose linear part is the coadjoint
representation. The author then proves that the moment map is equivariant with
respect to this affine action of $G$ on ${\mathcal{G}}^{*}$ and its action on
$M$ as a dynamical group of this manifold. The orbits of this affine action
are submanifolds embedded in ${\mathcal{G}}^{*}$ (in author’s meaning, most of
the geometers would say rather _immersed_) and are endowed with a symplectic
form whose expression involves the cocycle of $\mathcal{G}$, that is the
differential of $\theta$ at the neutral element. Nowadays this important
result is expressed by saying that ${\mathcal{G}}^{*}$ possesses a _Poisson
structure_ whose _symplectic leaves_ are the orbits of the affine action for
which the moment map is equivariant, that this Poisson structure remains
unchanged under this affine action and that the moment map is a _Poisson map_.
When this Poisson structure on ${\mathcal{G}}^{*}$ was discovered by the
Norwegian mathematician Sophus Lie (1842–1899) in the special case in which
the cocycle $\theta$ vanishes. In the general case, it was rediscovered
independantly by Alexander Kirillov, Bertram Kostant and the author, Jean-
Marie Souriau. In his book _Structure des systèmes dynamiques_ , he does not
use the concept of Poisson structure, rather he uses the fact that the orbits
of the affine action of $G$ on ${\mathcal{G}}^{*}$ are symplectic manifolds,
which he calls _symplectic manifolds defined by a Lie group_. He shows that
when the dynamical group $G$ of a symplectic manifold $(M,\sigma)$ is
connected, and when its action is transitive and possesses a moment map
$\Psi$, this map is a local symplectomorphism of $M$ onto an orbit of the
affine action of $G$ on ${\mathcal{G}}^{*}$ for which $\Psi$ is equivariant.
The map $\Psi$ is a symplectomorphism if and only if, moreover, the isotropy
group of a point of $M$ is connected. He presents some examples of this
important result, often called nowadays _Kostant-Souriau theorem_.

#### Chapter III, Mechanics.

Relatively long (105 pages), this chapter begins with the study of a
mechanical system composed of material points in a fixed Galilean frame. The
author writes _Newton’s equation_ expressing the equality of the force acting
on each of those points and the product of the mass by its acceleration. He
treats briefly the case of a unique material point placed in a Coulomb field
(Kepler’s problem), next the _$N$ -body problem_ of celestial mechanics. He
introduces then the notion of _constraint_ , presents the _principle of
virtual work_ and the conditions in which a constraint is called _ideal_. He
uses, to study the motion of a rigid body, the _group of_ _Euclidean
displacements_ and establishes the equations of motion.

Returning to the case of a system of $N$ material points without constraints,
subjected to forces expressed by differentiable functions of the time and of
the positions and velocities of these points, the author shows that the
equations of motion are expressed as the differential equation associated to a
vector field depending on time, defined on the _evolution space_ of the system
( a set, isomorphic to $\mathbb{R}^{6N+1}$, made of multiplets composed of the
time and of the positions and velocities of the $N$ material points). These
equations determine a _foliation in curves_ of the evolution space. Each of
these curves is the mathematical expression of a possible motion of the
system. It is why the author calls _space of motions_ of the system the set
made of these curves. He proves that the space of motions is a differential
manifold (not always Hausdorff) of dimension $6N$ and there exists a natural
projection, smooth and anywhere of rank $6N$, of the evolution space on the
space of motions. Then he proves the existence, on the evolution space of the
system, of a remarkable $2$-form $\sigma$, which he calls _Lagrange’s form_ ,
because it was used in 1808 by Lagrange in his works in celestial mechanics.
As pointed out by the author, this form was used in mechanics around 1950 by
the French mathematician François Gallissot [4]. For a unique material point
of mass $m$ whose position and velocity vectors are, respectively, ${\bf r}$
and ${\bf v}$, on which acts a force ${\bf F}$, the components of these three
vectors in a fixed orthonormal frame of the space being, respectively,
$(r_{1},r_{2},r_{3})$, $(v_{1},v_{2},v_{3})$ and $(F_{1},F_{2},F_{3})$, this
form reads

$\sigma=\sum_{i=1}^{3}(m\mathrm{d}v_{i}-F_{i}\mathrm{d}t)\wedge(\mathrm{d}r_{i}-v_{i}\mathrm{d}t)\,,\
\hbox{that can be written}\ \sigma=(m\mathrm{d}{\bf v}-{\bf
F}\mathrm{d}t)\dot{\wedge}(\mathrm{d}{\bf r}-{\bf v}\mathrm{d}t)\,,$

where the symbol $\dot{\wedge}$ denotes the operator combining the dot product
and the exterior product.

The Lagrange form of a system of $N$ material points is the sum of the
Lagrange forms of all points of the system. It determines the vector field of
which the system’s motions are the integral curves, since its kernel is the
sub-bundle that determines the foliation in curves of the evolution space. The
author remarks that it is always possible to choose $N$ differentiable vector
fields ${\bf B}_{j}$, defined on the evolution space, and to define $N$ other
vector fields ${\bf E}_{j}$ so that the force ${\bf F}_{j}$ acting on the
$j$-th material point is ${\bf F}_{j}={\bf E}_{j}-{\bf B}_{j}\times{\bf
v}_{j}$, where ${\bf v}_{j}$ is the velocity of this material point.

When changing the reference frame, the parameterization of the evolution space
and the expression of the Lagrange form are modified, particularly because
inertial forces must be included in the forces acting on the material points
(centrifugal force and Coriolis’ force). The author states the _principe of
Galilean relativity_ which claims the existence of preferential reference
frames, called _inertial reference frames_ , such that the expression of the
Lagrange form of an _isolated_ system is the same in all inertial reference
frames. He shows that the relative motion of an inertial reference frame with
respect to another one is a translation motion at a constant velocity. The set
of all changes of inertial reference frames is a Lie group, called the
_Galilei group_. The author gives a matrix expression of this group, which is
of dimension $10$. It acts on the evolution space of an unconstrained system
of material points by an action which preserves the Lagrange form. Using the
principle of virtual works, the author extends the notions of evolution space,
Lagrange form and space of motions, for a system of $N$ material points
involving ideal constraints, which can be either holonomic or non holonomic
with a linear dependence on the velocities. The results obtained for the
unconstrained systems of material points remain valid, provided that the
constraints remain the same in all inertial reference frames.

The author calls _Maxwell’s principle_ the hypothesis that the exterior
derivative of the Lagrange $2$-form of a general dynamical system (not
necessarily made of material points) vanishes. For a system of material
points, since the vector fields ${\bf B}_{j}$ which appear in the forces ${\bf
F}_{j}={\bf E}_{j}-{\bf B}_{j}\times{\bf v}_{j}$ can be freely chosen, the
author formulates Maxwell’s principle in the following form: the vector fields
${\bf B}_{j}$ can be chosen in such a way that the Lagrange form is closed.
This condition determines these fields in a unique manner. The author proves
that, as a consequence of this principle, the vector fields ${\bf E}_{j}$ and
${\bf B}_{j}$ must not depend on the velocities of the material points and
must verify both _Maxwell equations_

$\mathop{\rm rot}\nolimits{\bf E}_{j}+\frac{\partial{\bf B}_{j}}{\partial
t}=0\,,\quad\mathop{\rm div}\nolimits{\bf B}_{j}=0\,,\quad 1\leq j\leq N\,.$

The author shows that Maxwell’s principle is well verified in a lot of cases:
the $N$-body problem, a material point in the gravity field, an electrically
charged particle in an external electromagnetic field. In this latter case,
the above equations are the first two Maxwell’s equations (Maxwell-Faraday
equation and Maxwell-Thomson equation) that must be verified by the external
electromagnetic field. The author deduces from them the well-known formula
giving the expression of the _Laplace force_ , and concludes that this force
_is not_ a relativistic effect since it results from the application of
Maxwell’s principle in the framework of classical mechanics.

By contrast, the creation of an electromagnetic field by electric charges in
motion, mathematically described by the last two Maxwell equations (Maxwell-
Ampère equation and Maxwell-Gauss equation) are relativistic effects which do
not appear in the framework of classical mechanics. The author adopts
Maxwell’s principle as a new _law of the mechanics_. This allows him the
study, in the framework of classical mechanics, systems more general than
those made of material points. For the systems of material points, Maxwell’s
principle allows, under certain conditions, to define a Lagrangian and to show
that the Lagrange form is nothing else than the exterior derivative of the
Cartan form defined in the first chapter, in the study of calculus of
variations. We can then apply the _principle of least action_ , often
considered as an essential piece of analytical mechanics. Moreover, when the
Lagrangian is hyper-regular, one can associate to it a Hamiltonian and use for
the study of motions the Lagrangian and Hamiltonian formalisms. Without
denying the importance of the principle of least action nor the usefulness of
these formalisms, the author declares that these concepts seem to him less
fundamental than Maxwell’s principle. His viewpoint seems to him justified
because the existence of a Lagrangian is ensured only locally, and because
there exist important systems, such as those made of particles with spin, to
which Maxwell’s principle applies while they have not a globally defined
Lagrangian. In the sequel, the author will not use the principle of least
action except to present, very briefly, the _method of variation of constants_
introduced by Lagrange in 1809 during his works on the slow variations of the
orbital elements of planets.

The Lagrange form projects onto the manifold of motions, and its projection is
a closed form, which is symplectic since it is automatically non degenerate.
When the system is isolated, the Galilei group acts on the evolution space and
on the space of motions. The moment map of this action on the evolution space
is a first integral of the motion. The author details its ten components,
which can be regrouped in three vectors $\bf p$, $\bf l$, $\bf g$ and a scalar
$E$. He gives their physical meaning: $\bf p$ is the total linear momentum;
$\bf l$ is the total angular momentum; the equality ${\bf g}=$ Constant
conveys the fact that the center of mass moves on a straight line at constant
velocity; the scalar $E$, defined modulo an additive constant, is the total
energy.

Using a result due to V. Bargmann [5], who proved that the symplectic
cohomology of the Galilei group is of dimension $1$, the author shows that the
class of symplectic cohomology of the action of the Galilei group on the space
of motions of an isolated system made of material points in interaction can be
interpreted as the _total mass_. The writing of the equations of motion gives
two noteworthy results: the vector fields ${\bf B}_{j}$ necessarily all
vanish; the total force and the total torque of the interaction forces are
necessarily zero. The first result shows that in classical mechanics, a system
made of material points cannot describe moving magnets. The second one
expresses the _principle of equality of action and reaction_ , which appears
as a consequence of Maxwell’s principle and of the principle of Galilean
relativity.

Next the author presents several properties of the dynamical groups of a
dynamical system and gives numerous examples. For the Kepler problem (motion
of a material point in a Coulomb field) he shows how to _regularize_ the
manifold of motions and explains the origin of the exceptional first integral,
often called the _Lenz vector_ or the _Laplace vector_ , which should rather
be called the _eccentricity vector_. Although the author does not say it, it
should be stressed that this first integral was discovered by the Swiss
mathematician Jakob Hermann (1678–1833) [6].

In the paragraph _The Principles of symplectic mechanics_ , the author first
works in the framework of non relativistic, classical mechanics. He no longer
is limited to systems of material points and adopts the three following
assertions as new axioms of mechanics:

  1. 1.

The space of the motions of a dynamical system is a _connected symplectic
manifold_.

  2. 2.

If several dynamical systems evolve independently, the manifold of motions of
the composite system is the _symplectic direct product_ of the spaces of
motions of the component systems.

  3. 3.

If a dynamical system is isolated, its manifold of motions admits the _Galilei
group_ as a dynamical group.

It is an _extension_ of the principles generally admitted in classical
mechanics, which will allow the author to consider new dynamical systems
having a physical interest.

Since, for an isolated system, the author identifies the _mass_ of the system
with the number $m$ which marks the class of cohomology of the Galilei group
action on the space of motions, he can now consider systems of positive, null
or negative mass. He keeps, for the more general systems that he will
consider, the physical interpretation of the components of the Galilean moment
which he has previously given: the vectors ${\bf l}$ and ${\bf p}$ are,
respectively, the _angular momentum_ and the _linear momentum_ of the system,
the scalar $E$ is its _energy_. As for ${\bf g}$, it allows, for a system of
non vanishing mass $m$, to define the _center of mass_ of the system that the
author calls the _center of gravity_ or the _barycenter_ : it is the point
whose position vector, at each instant $t$, is ${\bf R}=({\bf p}t+{\bf g})/m$.
Hence this point moves on a straight line at constant velocity ${\bf p}/m$.
The author chooses as fundamental quantities the length $L$, the time $T$ and
the action $A$, and indicates the dimensionnal equations of the encountered
quantities in the mathematical description of the dynamical system:
coordinates of an element of the evolution space, Lagrange form, components of
the matrices of an element of the Galilei group, of an element of its Lie
algebra and of an element of its dual.

Next, the author proves two theorems concerning the action of a dynamical
system on a symplectic manifold, that he might have placed in Chapter II. The
first provides, by means of the action of a dynamical group $G$ on a connected
symplectic manifold $V$ and the associated symplectic cocycle, the expressions
of a moment and that of the associated symplectic cocycle of the restriction
of this operation to an invariant subgroup $G^{\prime}$ of $G$. The second
applies to a moment $\psi$ of the action of a connected Abelian dynamical
group $G$ on a connected symplectic manifold $V$ when the differential $f$ at
the identity element of the symplectic cocycle $\theta$ associated to this
moment is a non degenerate bilinear form on the Lie algebra $\mathcal{G}$ of
$G$.

The author points out that there exist two Abelian subgroups of dimension $3$
of the Galilei group of changes of inertial reference frame of space-time: the
group of translations of the the frame of space, the frame of time remaining
unchanged, and the group of changes of inertial reference frame of space-time
in which a first inertial frame is replaced by a second one whose relative
motion with respect to the first one is a translation at a constant velocity,
the time frame remaining once again unchanged. The direct product of these two
groups is a normal subgroup of the Galilei group, isomorphic to the additive
group $\mathbb{R}^{6}$.

Applying the above-mentioned theorems, the author shows that the space of
motions of an isolated system of mass $m\neq 0$ is the direct product of two
spaces of motions: the _space of motions of the center of mass_ , isomorphic
to the space of motions of a material point of mass $m$, and the _space of
motions around the center of mass_ , in which the center of mass remains
always at the origin of the reference frame of space. This important result,
called the _barycentric decomposition_ of the motions of an isolated system,
is well known in classical mechanics. The Galilei group is a dynamical group
both of the space of motions of the center of mass and of the space of motions
around the center of mass, but acts on the last one only through the quotient
by its normal Abelian subgroup isomorphic to $\mathbb{R}^{6}$. This quotient
is isomorphic to $\mathrm{SO}(3)\times\mathbb{R}$. Hence the direct product of
the Galilei group and of $\mathrm{SO}(3)\times\mathbb{R}$ is a dynamical group
of any isolated system of non vanishing mass. The author gives several
examples including the one, non classical, of a particle with spin.

Next the author tackles the study of the relativistic systems. He calls
_Lorentz frame_ an inertial reference frame of space-time in which the time
unit has been chosen in such a way that the velocity of light is equal to $1$.
He presents the essential mathematical concepts used in the special theory of
relativity : the Minkowski space, the Lorentz group and the Poincaré group. He
gives the expressions of a diffeomorphism of
$\mathrm{SO}(3)\times\mathbb{R}^{3}$ onto the restricted Lorentz group ( the
connected component of this group containing the identity) and of matrix
representations of the Lie algebras of these groups. He points out, without
proof, several important properties: the Lie algebra of the Poincaré group is
equal to its derived algebra; the symplectic cohomology of the Poincaré group,
and more generally its cohomology valued in the dual of its Lie algebra, are
trivial.

In special relativity, the passage from a Lorentz frame to another one is made
by the action of an element of the restricted Poincaré group. The author
establishes the formula of change of Lorentz frames and gives its matrix
expression. The firts two axioms of symplectic mechanics remain unchanged, and
in the third one, the Galilei group is replaced by the restricted Poincaré
group. Therefore these axions are

  1. 1.

The space of motions of a dynamical system is a _connected symplectic
manifold_.

  2. 2.

If several dynamical systems evolve independently, the manifold of motions of
the composite system is the _direct product_ of the spaces of motions of the
component systems.

  3. 3.

If a dynamical system is isolated, its manifold of motions admits the
_restricted Poincaré group_ as a dynamical group.

It is always possible to choose the moment map of the action of the restricted
Poincaré group on the space of motions of an isolated relativistic dynamical
system so that the cocycle associated to the moment map is null. This
condition determines the moment map in an unique way, while in classical
mechanics, the moment map of the action of the Galilei group on the space of
motions of an isolated system depends on an arbitrary additive constant.
Moreover, in relativistic mechanics, the barycentric decomposition of motions
of an isolated system no longer exists.

The intersection of the Galilei and Poincaré groups, considered as subgroups
of the group of affine transformations of space-time, is a dynamical group of
dimension $7$ of isolated dynamical systems, both classical and relativistic.
The moment map of its action on the space of motions of an isolated system is
composed of the vectors ${\bf l}$, ${\bf p}$ and the scalar $E$, that the
author interpreted in classical mechanics as being the angular momentum, the
linear momentum and the energy. For a relativistic system, the author chooses
to conserve for these quantities the same interpretation as in classical
mechanics. The choice of a Lorentz frame allows to associate to the couple
$({\bf p},E)$ a vector $P$ of the Minkowski space-time, called the _$4$
-momentum_ or the _energy-momentum vector_ of the system. Next the author
indicates some formulae useful in geometry of oriented Minkowski space,
refering for more details to his book [7]. The action of the Poincaré group on
the space of motions of an isolated relativistic system reveals a second
vector $W$ of the Minkowski space-time, called the _polarization vector_ ,
orthogonal to the energy-momentum vector $P$.

In a long section, the author proposes a mechanistic description of elementary
particles. In the framework of relativistic mechanics, an isolated dynamical
system is said to be _elementary_ when the Poincaré group acts transitively on
the space of its motions. The moment map of its action is then a symplectic
diffeomorphism of this space onto a coadjoint orbit of the Poincaré group. For
the author, the so defined _elementary systems_ are mathematical models for
_elementary particles_ of physicists. He uses the type (timelike, spacelike or
lightlike, or in other words isotropic) of the quadrivectors $P$ and $W$,
defined in the previous section, for a classification of elementary systems.
By these means, he obtains a large part of the physicists’ classification of
elementary particles. Below, briefly summarized, his resultss are presented.

Case 1, a particle with spin. It is when $P$ is timelike and when $W$ (which,
being orthogonal to $P$, is spacelike) is non-zero. The author proves that two
real numbers $m$ and $s$, whose expressions are given in terms of $P$ and $W$,
can be interpreted as the _mass_ and the _spin_ of the particle. These numbers
are constant on the space of motions of the system. The mass $m$ is non-zero,
but can be either positive or negative, while the spin $s$ is always strictly
positive. For each given pair $(m,s)$, with $m\neq 0$ and $s>0$, there exists
only one model of particle with mass $m$ and spin $s$. Its space of motions is
$8$-dimensional. For each motion of the system, there exists an affine
straight line of the Minkowski space-time, parallel to the timelike
quadrivector $P$, interpreted as the _trajectory_ of the particle. By
expressing the energy-momentum quadrivector $P$ in any Lorentz reference
frame, the author observes that the norm of the velocity $v$ of the particle
in that reference frame is always smaller than $1$ (which is, with the chosen
units, the norm of the velocity of light) and obtains the famous Einstein’s
formula

$E=\frac{m}{\sqrt{1-\|{\bf v}\|^{2}}}\,.$

Case 2, a particle without spin. It is when $P$ is timelike and $W=0$. There
still exists a real number $m\neq 0$, expressed in terms of $P$, interpreted
as the _mass_ of the particle, constant on the space of motions of the system.
For each given real $m\neq 0$, there exists only one model of particle without
spin with mass $m$, and its space of motions is $6$-dimensional. Still in that
case, for each motion of the system, there exists an affine straight line of
the Minkowski space-time, parallel to the timelike quadrivector $P$,
interpreted as the _trajectory_ of the particle. In any Lorentz reference
frame the norm of the velocityof the particle is always smaller than $1$, the
energy $E$ and the velocity $v$ of the particle are related by the above
written Einstein’s formula.

Case 3, a massless particle. It is when both $P$ and $W$ are non-zero and
lightlike. The author defines three real numbers $\eta=\pm 1$, $\chi=\pm 1$
and $s>0$, interpreted, respectively, as the _sign of the energy_ , the
_helicity_ and the _spin_ of the particle, expressed in terms of $P$ and $W$,
constant on the space of motions of the system. For each triple
$(\eta,\chi,s)$ of real numbers satisfying $\eta=\pm 1$, $\chi=\pm 1$ and
$s>0$, there exists only one model of massless particle with sign of the
energy $\eta$, helicity $\chi$ and spin $s$, and its manifold of motions is
$6$-dimensional. For each motion of the system, there exists a three-
dimensional affine subspace of the Minkowski space-time which, interpreted in
any Lorentz reference frame, can be described as a two-dimensional spacelike
plane moving at the velocity of light, called the _wavefront_ of the particle.
There is no more an affine straight line of the Minkowski space-time which can
be considered as the trajectory of the particle. More precisely, for each
Lorentz reference frame, there is such an affine straight line, which is
lightlike and contained in the wavefront of the particle. However, this
straight line depends on the chosen Lorentz reference frame and sweeps the
whole wavefront when the considered Lorentz reference frame takes all the
possible values.

The author briefly indicates the existence of other elementary systems, for
example _tachyons_ , which do not correspond to known elementary particles.
Then he looks at _non relativistic elementary particles_ , beginning by
particles without spin. He obtains a mathematical model of such particles by
means of a suitable change of variables in which the velocity of light $c$
appears, and then by letting $c\mapsto+\infty$. The model so obtained
corresponds to material points considered in classical non-relativistic
mechanics. The same procedure starting with relativistic particles with spin
leads to a model of non-relativistic material point with spin, interpreted as
a _proper angular momentum_. He briefly indicates another way in which
mathematical models of non-relativitic particles could be obtained, using as
spaces of motions orbits of affine actions of the Galilei group on the dual of
its Lie algebra, which may involve a symplectic cocycle. By this means he
obtains models of non-relativistic massless particles moving at an infinite
velocity, each model being characterized by three real numbers $\chi=\pm 1$,
$s>0$ and $k>0$, called, respectively, the _helicity_ , the _spin_ and the
_color_ of the considered particle.

At the end of this section, the author explains that the theory of general
relativity argues in favour of physical elementary dynamical systems whose
space of motions admits the full Poincaré group $G^{\prime}$ as a dynamical
group. Such a system’s space of motions may have several connected components.
The full Poincaré group $G^{\prime}$ has four connected components, and all
its elements are obtained by composition of an element of the restricted
Poincaré group $G$ (the connected component of the neutral element) with
elements of two discrete subgroups, each with two elements: the group of
_space inversions_ (exchange of right and left), and the group of _time
reversals_ (exchange of past and future). The author fully discusses geometric
properties of the moment map of a Hamiltonian action of $G^{\prime}$ on a
(maybe non-connected) symplectic manifold, as well as geometric properties of
its coadjoint orbits, and presents their consequences for physical isolated
elementary systems whose space of motions admits $G^{\prime}$ as a dynamical
group. He considers elementary particles first with a non-zero mass, then with
zero mass. Known massless particles (photons and neutrinos) exist with two
opposite helicities, and the author considers this fact as an argument in
favour of the admission of the group of space inversions as an invariance
group of mechanics.

Chapter III ends with a study of particles dynamics first in the framework of
classical mechanics in a fixed inertial reference frame. Starting from the
dynamics of a free material point, the author explains how to describe the
dynamics of an electrically charged particle submitted to an electric fild
$\bf E$ and a magnetic field $\bf B$. He uses a modification of the Lagrange
form to account for the effets of $\bf E$, $\bf B$ and the electric charge of
the particle. Using the fact that $\bf E$ and $\bf B$ satisfy the Maxwell
equations, he proves that this modified Lagrange form still is closed, and
that its kernel, of dimension $1$, still determines a foliation in curves of
the evolution space, and derives the equations of motion. The same procedure,
in which the symplectic form of the space of free motions of a relativistic
particle is used (instead of that of the space of free motions of a classical
particle), leads to the equations of motion of a relativistic electrically
charged particle submitted to an electromagnetic field. In these equations, in
good agreement with experimental results, appears the relativistic correction
of the linear momentum well known by physicists.

For a particle with spin, either classical or relativistic, submitted to an
electromagnetic field, the author explains that experiments show that still
another term must be added to the Lagrange form. This term is the product by a
constant $\mu$ (later interpreted as the module of the magnetic moment of the
particle) of the exterior derivative of a $1$-form $\varpi$ which, at the non-
relativistic limit, is the product of $\mathrm{d}t$ by a function. The
equations of motion so obtained in the non-relativistic approximation are in
good agreement with the Stern and Gerlach experiments, as well as with the
precession of spin and the magnetic resonance phenomena. For a relativistic
particle these equations become very complicated. When the electromagnetic
field is constant, they are used for the measurement of the _anomalous
magnetic moment_ of particles.

The author now considers $n$ non-interacting particles, which may be either
free, or subjected to a field. Let $U_{i}$ be the space of motions of the
system made by the $i$-th particle alone. The space of motions $U$ of the
system is the direct product of the spaces of motions $U_{i}$, $1\leq i\leq
n$. The author describes two different ways in which an evolution space can be
built for the system, which lead to two different evolution spaces,
respectively called the _synchronous evolution space_ and the _asynchronous
evolution space_. The synchronous evolution space is obtained by adding to the
space of motions $U$ one dimension for the time, and by taking the initial
conditions of all particles at the same time. Its dimension is $\dim U+1$. The
asynchronous evolution space, built by taking the initial conditions of the
particles at different times, is of dimension $\dim U+n$, and involves $n$
different times, one for each particle. Both the synchronous and the
asynchronous evolution spaces are presymplectic manifolds which project onto
the space of motions $U$. The synchronous evolution space should be used only
for non-relativistic particles since it involves a notion of simultaneity,
which in relativistic physics depends on the choice of a Lorentz reference
frame. When all the particles of the system are identical and submitted to the
same field, the spaces of motions $U_{i}$, for all $i\in\\{1,\ldots,n\\}$, are
equal to $U_{1}$ and one could think that the space of motions of the system
is $(U_{1})^{n}$. However, experiments have shown that motions which only
differ by the labelling of particles should be identified. The true space of
motions of the system is the set of equivalence classes of $n$-uples
$(x_{1},\ldots,x_{n})\in(U_{1})^{n}$ which satisfy $x_{i}\neq x_{j}$ for
$i\neq j$, $1\leq i,j\leq n$, two $n$-uples $(x_{1},\ldots,x_{n})$ and
$(x^{\prime}_{1},\ldots,x^{\prime}_{n})$ being equivalent if there exists a
permutation $\sigma$ of $\\{1,\ldots,n\\}$ such that
$x^{\prime}_{i}=x_{\sigma(i)}$ for all $i$, $1\leq i\leq n$. The author
indicates the expression of the Lagrange form and remarks that the use of a
set of equivalence classes as a set of motions of the system is a consequence
of the indiscernability of particles which does not involve quantum mechanics.

A “classical” method for obtaining the equations of motion of a system of
interacting particles uses the synchronous evolution space of the system when
the particles do not interact, with a Lagrange form modified by addition of a
suitable term involving an _interaction potential_. This method works
successfully for celestial mechanics. The author uses it for a system of non-
relativistic particles with spin, indicates the equations of motion and, when
the system is isolated, writes the formulae which express the constancy of the
Galilean moment map. The equations so obtained take into account the
electrostatic and magnetostatic forces exerted by each moving particle on the
other particles. However, they do not account for other small, but measurable
relativistic effects, such as the Laplace force. To account for these effects,
the author considers the use of the asynchronous evolution space. He proves
that the use of that space would prohibit the existence of interactions
between the particles, and concludes that a way to get around this difficulty
could be to abandon the idea of localized particles, the distinction between
particles and fields being a non-relativistic approximation.

The _scattering theory_ is an approximate mathematical description of a system
of interacting particles in which it is assumed that when each particle is far
enough from all other particles, the interaction forces can be neglected. The
author presents a mathematically rigorous version of this theory. He assumes
that unscattered and scattered dynamical systems share the same evolution
space with two different Lagrange forms, equal of an open subset $\Omega$ of
the evolution space. The complement of $\Omega$, on which the Lagrange forms
of the scattered and unscattered systems are not equal, is called the
_scattering source_. A scattered motion is said to be _constrained_ when it is
wholly contained in the scattering source. The author looks at unconstrained
scattered motions contained in the scattering source only for a bounded time
interval. He states, without a complete proof, a theorem according to which
such a motion coincides with two different unscattered motions, one before it
enters the scattering source and another one after it has finally left the
scattering source. The so established correspondence between two unscattered
motions is a symplectomorphism between two open subsets of the set of motions
of the unscattered system. The author defines the _symmetry group_ of the
scattering source and proves that it is a dynamical group for both systems,
scattered as well as unscattered. For _bounded_ , _static_ or _conservative_
scattering sources, some properties of this group can be deduced. The author
briefly presents the dynamical system made by photons in a refracting
telescope. Due to a term containing a length (later interpreted as the
wavelenth of the light) in the formulae he obtains, the image of a distant
star can never be a point. When this term is neglected, one obtains the
_geometrical optics_ approximation. Using the scattering theory, the author
discusses, in relativitic physics, the reflection of light on a moving mirror
and obtains formulae for the _Doppler effect due to reflection_.

Collisions of relativistic free particles with non-zero masses are finally
discussed by the author, with the help of the scattering theory. Though no
valid model of relativistic interacting particles is available, the author
obtains some properties of the symplectomorphism which relates the free
motions of the particles before and after their collision. He proves that this
symplectomorphism commutes with the action of elements of the Poincaré group,
which implies that the total momentum and the total energy of the system of
particles are conserved by collisions. As a conclusion, he states that knowing
this symplectomorphism could allow to study the _constrained motions_ by the
technique of analytic continuation.

#### Chapter IV, Statistical mechanics.

It contains two sections. The first one, of about 50 pages, is essentially
mathematical. The second one, of about 35 pages, presents the principles of
statistical mechanics.

The first section of this chapter begins by introducing various concepts
related to smooth manifolds, topological spaces, ordered vector spaces (called
_Riesz spaces_), normed vector spaces, specially Banach and Hilbert spaces. A
very condensed course (about 30 pages) in Measure Theory and Integration
follows, with some notions in Probability. The author uses the presentation,
privileged by the Boubaki school, in which a measure on a smooth Hausdorff
manifold $V$ is an element of the topological dual vector space of the space
of continuous, compactly supported functions on $V$, instead of defining first
measurable parts of $V$, and then a measure as a denombrably additive function
defined on the set of measurable parts. Then the author defines _probability
measures_. A measure $\lambda$ on $V$ is said to be _defined by an everywhere
positive continuous field of densities_ on $V$, when its expression in every
chart is the product, by an everywhere strictly positive continuous function,
of the Lebesgue measure. The measure $\lambda$ being fixed, the author
considers a probability measure whose density with respect to $\lambda$ is a
continuous function $\rho\geq 0$. He defines the $\lambda$-_entropy_ of this
probability measure by setting

$s_{\lambda}(\rho)=\begin{cases}\displaystyle\int_{V}-\rho(x)\log\bigl{(}\rho(x)\bigr{)}\lambda(\mathrm{d}x)&\text{if
this integral converges,}\\\ \displaystyle-\infty&\text{if the above integral
does not converge\,.}\end{cases}$

By convention, at points $x\in V$ where $\rho(x)=0$, the value taken by the
function $x\mapsto-\rho(x)\log\bigl{(}\rho(x)\bigr{)}$ is $0$.

A continuous map $\Psi$, defined on $V$, with values in a finite-dimensional
vector space $E$ being given, the author calls _generalized Gibbs probability
law_ any completely continuous probability law for which the map $\Psi$ is
integrable, whose density $\rho$ with respect to $\lambda$ is expressed as

$\rho(x)=\exp\Bigl{(}-\bigl{(}z+\langle
Z,\Psi(x)\rangle\bigr{)}\Bigr{)}\,,\quad\hbox{with}\ x\in V\,.$

In the above equality, $Z$ is an element of the dual vector space $E^{*}$ of
$E$, which must be such that the integrals below, appearing on the right hand
sides of the equalities defining $I_{0}$ and $I_{1}$, are convergent. The real
$z$ must be chosen in such a way that the integral of $\rho$ on $V$ with
respect to the measure $\lambda$ is $1$.

The _normal distribution_ on $\mathbb{R}^{n}$ is a generalized Gibbs
probability law for a suitable choice of the map $\Psi$.

The mean value of $\Psi$, for the probability law of density $\rho$ with
respect to $\lambda$, being denoted by $M$, and setting

$I_{0}=\int_{V}\exp\Bigl{(}-\langle
Z,\Psi(x)\rangle\Bigr{)}\lambda(\mathrm{d}x)\,,\quad
I_{1}=\int_{V}\Psi(x)\exp\Bigl{(}-\langle
Z,\Psi(x)\rangle\Bigr{)}\lambda(\mathrm{d}x)\,,$

the author can write

$z=\log(I_{0})\,,\quad
M=\int_{V}\Psi(x)\rho(x)\lambda(\mathrm{d}x)=\frac{I_{1}}{I_{0}}\,.$

The $\lambda$-entropy of the generalized Gibbs law of density $\rho$ with
respect to $\lambda$ is

$s(\rho)=z+\langle Z,M\rangle\,.$

The author proves that, on the set of completely continuous probability laws
for which the mean value of $\Psi$ is $M$, the $\lambda$-entropy functional
has a strict maximum at the generalized Gibbs probability law of density
$\rho$ with respect to $\lambda$. Moreover, when the set of values of $\Psi$
is not contained in an affine subspace of $E$ of dimension strictly smaller
than $\dim E$, he proves that the map $Z\mapsto M$ is injective. He derives
conditions for the differentiability with respect to $Z$, under the sign
$\int$, of the integrals $I_{0}$ and $I_{1}$. He will later improve these
results in his paper [8], published a few years after his book [1].

The author then assumes that the Hausdorff manifold $V$ is endowed with a
symplectic form and that a connected Lie group $G$ acts on it by a Hamiltonian
action. For $\lambda$, he chooses the Liouville measure, and for the map
$\Psi$ a moment map of the action of $G$. The vector space $E$ is therefore
now the dual vector space ${\mathfrak{g}}^{*}$ of the Lie algebra
$\mathfrak{g}$ of $G$, and its dual $E^{*}$ is ${\mathfrak{g}}$. He denotes by
$\Omega$ the largest open subset of $\mathfrak{g}$ on which the integrals
$I_{0}$ and $I_{1}$ are convergent and define differentiable functions of the
variable $Z$ whose differentials are continuous and can be obtained by
differentiation under the $\int$ sign. A generalized Gibbs probability law can
then be associated to each $Z\in\Omega$. The author indicates the
corresponding expressions of $z=\log(I_{0})$, of the entropy $s$ and of the
mean value $M$ of the moment map, considered as functions of the variable
$Z\in{\mathfrak{g}}$. He proves that when $G$ acts effectively on $V$, the map
$Z\mapsto M$ is injective and open, therefore is a diffeomorphism of $\Omega$
onto an open subset $\Omega^{*}$ of ${\mathfrak{g}}^{*}$. By using the inverse
map, $z$, $s$ and $Z$ can be considered as functions of the variable $M$,
which spans the open subset $\Omega^{*}$ of ${\mathfrak{g}}^{*}$. The author
proves that these functions can be differentiated and indicates the
expressions of their differentials. Adding a constant to the moment map does
not change the generalized Gibbs laws on the manifold $V$. Their set is called
by the author the _Gibbs set_ of the considered Lie group action. Moreover the
author proves that the open subset $\Omega$ of $\mathfrak{g}$ is a union of
orbits of the adjoint action of $G$ and is endowed with a definite negative
Euclidean metric.

Results obtained by the author in the first section of Chapter IV are, in the
second section, applied to dynamical systems encountered in physics. A
_statistical state_ of such a system is a probability law on the space of its
motions. The author explains that in the _kinetic theory of gases_ , a gas in
a container at rest in a Galilean reference frame is considered as an assembly
of very small material particles whose motions are governed by the laws of
classical mechanics, interacting by instantaneous elastic collisions between
themselves and with the walls of the container. With these assumptions, the
gas can be modelled by a conservative Hamiltonian system. The one-dimensional
group of time translations acts on the symplectic manifold of motions of the
system by a Hamiltonian action, with the energy as a moment map. The author
explains that the entropy of the system increases with time, so assuming that
the _natural equilibria_ of the gas are elements of the Gibbs set of the group
of time translations is a very reasonable assumption. Each Gibbs state is
determined by an element $Z$ of the one-dimensional Lie algebra of this group,
which is a way of measuring the _gas temperature_. Each Gibbs state is
unaffected by the action of the group of time translations, since the adjoint
action of this group on its Lie algebra is trivial. The author indicates the
dimension equations of quantities which determine a Gibbs state, or appear in
its description. He explains that when a compound system is the union of
several subsystems, the tensor product of natural equilibria of the subsystems
is a natural equilibrium of the compound system only when the element $Z$ of
the Lie algebra of the group of time translations which determine the natural
equilibria is the same for all subsystems, in other words only when the
temperatures of all the subsystems are equal. If this condition is not
satisfied, the compound system is in a state of natural equilibrium only when
the subsystems cannot exchange energy between them. As soon exchanges of
energy can occur, even when they are very tiny, the compound system is no more
in a natural state of equilibrium.

For the determination of the Gibbs states of a gas, one has to calculate the
integral $I_{0}$. The _ideal gas approximation_ is when, for this calculation,
the only motions taken into account are those for which, at the considered
time, no collisions occur between particles or between a particle and the
walls of the container, and when the total volume of the particles is
considered as negligible in comparison with the volume of the container. The
author successively considers monatomic and polyatomic ideal gases and
derives, for a Gibbs state, the probability distribution of velocities of the
particles, which was determined by Maxwell in 1860. He explains the principle
of an _ideal gas thermometer_ used for precise measurements of the
temperature. He is led to the formula $Z=1/(kT)$, which expresses the absolute
temperature $T$ in terms of the element $Z$ of the Lie algebra of the group of
time translations, and of _Boltzmann’s constant_ $k$. He obtains the Mariotte,
Gay-Lussac and Avogadro laws for perfect gases well known by physicists. He
proves that for an ideal gas thermometer, the temperature is a random variable
whose probability law is very concentrated around its mean value, and
converges (for the weak topology) towards the Dirac measure when the number of
particles tends to infinity.

For a conservative system, the entropy $s$ of a Gibbs state is a smooth
function of the energy $E$ whose derivative is the element $Z$ of the Lie
algebra which determines the Gibbs state. Units being chosen, $Z$ becomes a
real number. Experimental results show that this number is always strictly
positive. The author can therefore consider the energy $E$ of a Gibbs state as
a function of its entropy $s$ and of other variables which describe the
system. For example, when the considered system is a gas, these additional
variables are the volume of the container, the number of particles, etc. The
author assumes that these variables can be globally described by an element
$u$ in a smooth Hausdorff manifold. Infinitesimal variations of a Gibbs state
are therefore described by the differential of $E$,

$\mathrm{d}E=\frac{\mathrm{d}s}{Z}+\varpi\mathrm{d}u=T\mathrm{d}S+\varpi\mathrm{d}u\,,$

where $\varpi$ denotes the partial differential of $E$ with respect to $u$,
and $S=ks$ ($k$ being Boltzmann’s constant). The author calls $S$ and $u$ the
_position variables_ , $T$ and $\varpi$ the associated _tension variables_.
For example, when the considered system is a gas in a container whose volume
may vary, the above expression becomes

$\mathrm{d}E=T\mathrm{d}S-p\mathrm{d}V\,,$

where $p$ is the _pressure_.

The author then explains that the infinitesimal variation of the energy $E$ is
the sum of the infinitesimal variations of two quantities $Q$ and $W$, called
respectively the _heat_ and the _work_ , which are not functions, but rather
_action integrals_ in the sense of calculus of variations. Variations of heat
and work depend on the path followed while the adjustable variables which
describe the state of the system changed. Different types of evolution of the
state of a system should be separately considered, for example _adiabatic
evolution_ , _isothermal evolution_ , etc.

The author defines the _heat capacities_ of a gas at _constant volume_ and at
_constant pressure_ , and indicates the expressions of the thermodynamic
functions of an _ideal fluid_. These expressions depend on the chosen model
for the particles. A table indicates their expressions and values for various
models (material point, particle with spin, etc). Agreement with experimental
results is good for a monoatomic gases, not so good for polyatomic gases and
for solids at low temperatures.

Since the group of time translations is a subgroup, but not a _normal_
subgroup, of the Galilei group, a dynamical system conservative in some
inertial reference frame is not conservative in a different inertial reference
frame. This important remark leads the author to introduce the new concept of
_covariant statistical mechanics_ by proposing the following principle:

_When a dynamical system is invariant by the action of some Lie subgroup
$G^{\prime}$ of the Galilei group, its natural equilibria are the elements of
the Gibbs set of the action of $G^{\prime}$._

Each natural equilibrium of such a system is determined by an element $Z$ of
the Lie algebra ${\mathfrak{g}}^{\prime}$ of $G^{\prime}$, which is a Lie
subalgebra of the Lie algebra of the Galilei group. The author observes that
$Z$ generalizes the inverse of the temperature, and discusses its physical
interpretation. For this purpose he chooses an inertial reference frame
${\mathcal{R}}_{0}$ of the Galilei space-time and considers the reference
frame ${\mathcal{R}}=\exp(tZ){\mathcal{R}}_{0}$, which generally is not
inertial. He looks at the indications given, in the reference frame
$\mathcal{R}$, by an ideal gas thermometer in equilibrium with the considered
system. It amounts to observe the system in a moving frame. The author proves
that it is as if $\mathcal{R}$ were an inertial frame and if the particles of
gas were submitted to additional forces (inertial forces, _i.e._ , centrifigal
and Coriolis forces). By this means he obtains general formulae for the
probability density of the Gibbs state associated to $Z$. He then discusses in
greater detail several examples: the wind (the relative motion of
$\mathcal{R}$ with respect to ${\mathcal{R}}_{0}$ is a translation at constant
velocity), accelerating rocket (this relative motion is a uniformly
accelerated translation), gas in a centrifuge (this relative motion is now a
rotation around an axis at a constant angular velocity). In this last example
he considers also a system made by particles with spin, and finds that the
most probable orientation of the particle’s spin is parallel to the rotation
axis.

One may wish to apply the above principle for a system invariant by the whole
Galilei group. However, the subset $\Omega$ of the Lie algebra of the Galilei
group made by elements which determine a Gibbs state is then empty. Looking at
the motions of the system around its center of mass, the author is led to
consider the just discussed equilibria in a rotating frame. For the author,
the rotation of celestial bodies observed in Astronomy confirms the validity
of his principle of covariant statistical mechanics.

For a relativistic dynamical system, the Galilei group must be replaced with
the Poincaré group. As above, the Gibbs state of the action of the whole
Poincaré group is empty, and the author is led to consider systems invariant
by the action of a subgroup $G^{\prime}$ of the Poincaré group. He presents in
more detail several examples: an ideal gas in a container at rest in an
inertial frame (in that example Maxwell’s distribution law for the velocities
of particles is slightly modified); relativistic wind; statistical equilibria
of photons. In this last example, the number of photons cannot be fixed. The
author explains how such a system can be described and even takes into account
the fact that photons can have two opposite circular polarizations. However,
his formula is not in agreement with the one obtained by Planck for the black-
body radiation, which is in very good agreement with experimental
observations. The author concludes that this last example must be dealt with
in the framework of quantum mechanics.

#### Chapter V, Geometric quantization.

It contains two long sections, each of around 40 pages. Before describing
their contents, let us recall that a _contact form_ on a smooth Hausdorff
manifold $Y$ is a differential $1$-form $\omega$, which nowhere vanishes on
$Y$, whose exterior derivative $\mathrm{d}\omega$ is non degenerate on
$\ker\omega$. The existence of a contact form on the manifold $Y$ has several
important consequences: $Y$ must be odd-dimensional, the $2$-form
$\mathrm{d}\omega$ is a _presymplectic form_ and there exists on $Y$ a unique
vector field $R_{Y}$, called the _Reeb vector field_ (in honour of the French
mathematician Georges Reeb), determined by the two equalities
$\mathrm{i}(R_{Y})(\mathrm{d}\omega)=0$ and $\mathrm{i}(R_{Y})\omega=1$.

The author defines a _quantum manifold_ as a smooth manifold $Y$ endowed with
a contact form $\omega$ such that all integral curves of the Reeb vector field
$R_{Y}$ are $2\pi$-periodic. These curves are then the orbits of an operation
on $Y$ of the one-dimensional torus. The set of these curves, in other words
the quotient of $Y$ by this operation, is a symplectic manifold
$(U,\sigma_{U})$, called by the author the _basis_ of the quantum manifold
$Y$. The quantum manifold projects on $U$, and the pull-back by the projection
map of the symplectic form $\sigma_{U}$ is the presymplectic form
$\mathrm{d}\omega$.

A _quantization_ of a given Hausdorff symplectic manifold $(U,\sigma_{U})$ is
defined by the author as the construction of a quantum manifold whose basis is
$(U,\sigma_{U})$. A symplectic manifold is said to be _quantizable_ when its
quantization is possible. The author indicates, without proof, a necessary and
sufficient condition in which a given symplectic manifold is quantizable: the
cohomology class of its symplectic form must be $2n\pi$, with $n$ an integer.
It is satisfied when $\sigma_{U}$ is the exterior derivative of a $1$-form.
The manifolds of motions of many mechanical systems, for example harmonic
oscillators, Kepler’s problem, and the $N$-body problem of celestial
mechanics, are therefore quantizable. The author proves that the manifold of
motions of a non-relativistic particle with spin is quantizable if and only if
the spin of the particle is integer or half integer, when expressed with
$\hbar$ as unit. The author indicates several examples of quantizable
symplectic manifolds, together with the full description of the corresponding
quantum manifolds: two-dimensional spheres of integer or half-integer radii,
spaces of motions of a relativistic particle, first with a non-zero mass, then
with zero mass, with spin $1/2$ or $1$.

Isomorphisms of quantum manifolds are called by the author _quantomorphisms_.
Any quantomorphism between two quantum manifolds projects onto a
symplectomorphism between their bases. Two quantizations $Y$ and $Y^{\prime}$
of the same symplectic manifold are said to be _equivalent_ when there exists
between them a quantomorphism which projects onto the identity map of their
common basis. A symplectic manifold is said to be _monoquantizable_ when all
its quantizations are equivalent.

A group $\Gamma$ of quantomorphisms of a quantum manifold $Y$ projects onto a
group of symplectomorphisms of its basis $U$, and its projection is a group
homomorphism. Conversely, a group $G$ of symplectomorphisms of the basis $U$
is said to be _liftable_ if there exists a group $\Gamma$ of quantomorphisms
of $Y$ which projects onto it. When in addition the projection of $\Gamma$
onto $G$ is a group isomorphism, $\Gamma$ is said to be an _isomorphic lift_
of $G$. The author proves that the set of isomorphic lifts of $G$ is in one to
one correspondence with the set of its characters (group homomorphisms of $G$
into the one-dimensional torus).

A theorem due to the author states that a simply connected quantizable
symplectic maniold is monoquantizable. Another theorem explains how to
quantize a symplectic manifold $U$ when a quantization of a covering manifold
$U^{\prime}$ of $U$ is known. Conversely, when a quantization $Y$ of $U$ is
known and when the group of symplectomorphisms of $Y$ determined by the
covering manifold $U^{\prime}$ isomorphically lifts onto a group $\Gamma$ of
quantomorphisms of $Y$, $\Gamma$ is a discrete group and can be used to build
a covering manifold $Y^{\prime}$ of $Y$, which quantizes $U^{\prime}$. Using
these two theorems, it can be proven that there exist as many non equivalent
quantizations of a quantizable connected symplectic manifold as its homotopy
group has distinct characters. Therefore, being simply connected, the two-
dimensional sphere, any symplectic vector space, the space of motions of a
free particle, (with or without spin, non-relativistic or relativistic), when
quantizable, are monoquantizable. The author then discusses the quantizability
of the manifold of motions of a system of $N$ particles without interactions.
If each of these particles can be distinguished from the others and has an
integer or half-integer spin, the space of motions of the system is
quantizable and simply connected, therefore monoquantizable. But if each
particle cannot be distinguished from the others, the space of motions of the
system has exactly two non-equivalent quantizations, corresponding to the two
distinct characters of the group of permutations of a set of $N$ elements.
These two quantifications are the Bose-Einstein and Fermi-Dirac quantizations,
well known to physicists.

A smooth vector field, defined on a quantum manifold $Y$, whose flow acts on
$Y$ by quantomorphisms, is called by the author an _infinitesimal
quantomorphism_. The Lie derivative of the contact form $\omega$ defined on
$Y$ with respect to an infinitesimal quantomorphism vanishes. The author
proves that each infinitesimal quantomorphism is associated to a smooth
function defined on the basis $U$ of $Y$. For example, the Reeb vector field
is an infinitesimal quantomorphism associated to the constant function whose
value at any point in $U$ is $1$. The Lie bracket of two infinitesimal
quantomorphisms is an infinitesimal quantomorphism, whose associated function
is the Poisson bracket of the functions associated to these two infinitesimal
quantomorphisms.

The author then discusses the _quantization_ of a dynamical group of a
quantizable symplectic manifold $U$, with the quantum manifold $Y$ as
quantization. When a Lie group acts on $Y$ by quantomorphisms, it acts also on
the basis $U$ by symplectomorphisms, the projection of $Y$ onto $U$ being
equivariant with respect to these actions. Therefore $G$ is a dynamical group
of $U$, and is said to be a _quantizable_ dynamical group. A quantizable
dynamical group of $U$ is always liftable. The author proves that when a
dynamical group of $U$ is quantizable, its symplectic cohomology is zero. He
gives examples which prove that this necessary condition is not sufficient.

A dynamical group of $U$ which is liftable, but not quantizable, may have an
extension which still is a dynamical group of $U$ and is quantizable. As a
first example, the author considers a symplectic vector space $E$ which acts
on itself by translations. It is a dynamical group of $E$. The product
$Y=E\times T$, where $T$ is the one-dimensional torus, is a quantization of
$E$, which happens to be a Lie group, called the _Weyl group_ , extension of
the Abelian group $E$. It acts on itself by translations, which are
quantomorphisms. The Weyl group is therefore a quantizable extension of the
Abelian group $E$, which itself is not quantizable since its symplectic
cohomology is not zero. As a second example, the author considers the group
$\mathrm{SO}(3)$ acting on a two-dimensionl sphere $S_{2}$ centered on the
origin of $\mathbb{R}^{3}$, endowed with its element of area form as
symplectic form. This symplectic manifold is quantizable if its total area is
an integer or an half-integer. In this latter case, the author proves that the
dynamical group $\mathrm{SO}(3)$, which is liftable since $S_{2}$ is simply
connected, is not quantizable, though its symplectic cohomomogy is zero. He
proves also that the group $\mathrm{SU}(2)$ is a quantizable extension of
$\mathrm{SO}(3)$. A similar situation occurs for the restricted Poincaré
group, which is a dynamical group of the manifold of motions of a relativistic
particle. This group is quantizable if and only if the particle’s spin is
integer. If the particle’s spin is half-integer, there exists a quantizable
extension of the restricted Poincaré group descibed by the author in terms of
_Dirac’s spinors_.

The author denotes by ${\mathcal{H}}(Y)$ the vector space of smooth, complex-
valued and compactly supported functions defined on a quantum manifold $Y$,
which are equivariant with respect to the one-dimensional torus $T$ actions on
$Y$ by the flow of the Reeb vector field and on $\mathbb{C}$ by multiplication
(the torus $T$ being identified with the set of complex numbers of modulus
$1$). After indicating the definitions and results in topology and functional
analysis he is going to use, he proves that ${\mathcal{H}}(Y)$ is a pre-
Hilbert space. He defines the _Hilbert space of the quantum manifold $Y$_ as
the completion of ${\mathcal{H}}(Y)$. He defines also $C^{*}$-algebras,
indicates some of their properties and proves that the set of bounded linear
endomorphisms of a Hilbert space is a $C^{*}$-algebra. He very shortly
presents many concepts about operators on a Hilbert space: self-adjoint,
normal, Hermitian, unitary operators, etc. For a given quantum manifold $Y$ of
basis $U$, the author then explains how to associate a Hermitian operator
$\widehat{\mathstrut u}$ on ${\mathcal{H}}(Y)$ to any smooth function $u$
defined on the basis $U$. He proves that the map $u\mapsto\widehat{\mathstrut
u}$ is linear and injective, that for $u=1$, $\widehat{\mathstrut u}$ is the
identity of ${\mathcal{H}}(Y)$, and that for any pair $(u,u^{\prime})$ of
smooth functions on $U$,

$\widehat{\mathstrut u}\circ\widehat{\mathstrut
u^{\prime}}-\widehat{\mathstrut u^{\prime}}\circ\widehat{\mathstrut
u}=-i\widehat{\mathstrut\\{u,u^{\prime}\\}}\,.$

He illustrates this important result in the case when $Y$ is the quantization
of a symplectic vector space, the smooth functions on this vector space being
canonical coordinates. The operators associated to smooth functions in this
way often are not bounded, which makes their study and use very difficult. He
proposes to build directly quantomorphisms acting on $Y$, without taking the
detour of infinitesimal quantomorphisms and exponentiation. He proves a
theorem which asserts the existence of an injective homomorphism of the group
of quantomorphisms of $Y$ into the group of unitary operators on the Hilbert
space of $Y$.

The second section of Chapter V begins with the statement and explanation of
the _Correspondence Principle_ , well known by physicist. Quantum mechanics is
not an autonomous theory: it cannot be formulated without using classical
mechanics. According to the Correspondence Principle, for each physical
phenomenon described in the framework of quantum mechanics, there should exist
a corresponding “classical approximation” described in the framework of
classical mechanics. For the author, this principle is an argument in favour
of the extension of classical mechanics he proposed in Chapter II, the
_Maxwell Principle_. Without this extension, the use of classical mechanics
would be limited to systems made of material points, excluding many phenomena
encountered in physics such as particles with spin.

The _quantization_ of a classical mechanical system is the construction of the
quantum mechanical system of which this classical system is an approximation.
In this section the author examines the possible application of the
mathematical theory of Geometric Quantization developed in the previous
section, to the quantization of real systems encountered in physics. His
initial assumption is: the space $U$ of classical motions of the system is a
_quantizable symplectic manifold_. The first consequence of this assumption
is: expressed with the constant $\hbar$ as unit, the value of the spin of
particles must be either an integer or a half-integer. This consequence is in
perfect agreement with experimental results.

Let the quantum manifold $Y$ be a quantization of the classical manifold of
motions $U$. A _state vector_ of $Y$ is any element $\Psi$ with norm $1$ of
the pre-Hilbert space ${\mathcal{H}}(Y)$. The author explains its
probabilistic interpretation: for each $\xi\in Y$, the real non-negative
number $|\Psi(\xi)|^{2}=\overline{\Psi(\xi)}\Psi(\xi)$ depends only on the
projection $x$ of $\xi$ on the manifold of motions $U$; the function so
defined on $U$ is the probability density, with respect to the Liouville
measure, of a probability law on $U$ which is the _statistical state_ of the
system (in the sense defined in Chapter IV) corresponding to the state vector
$\Psi$.

An _observable_ in classical mechanics is a smooth function $u$ defined on the
classical manifold of motions $U$. According to the British scientist P.A.M.
Dirac, _observables_ in quantum mechanics are Hermitian operators. In
agreement with this view, the author has proven in the previous section that a
Hermitian operator $\widehat{\mathstrut u}$ could be associated to the
classical observable $u$. This operator is the quantum observable of which $u$
is the classical approximation. When applied to the phase space of a classical
conservative system at a given time $t$, this idea leads to Dirac’s _quantum
equations of motion_ and to the _commutation relations_ chosen by the
physicists W. Pauli and W. Heisenberg as fundamental ingredients of quantum
mechanics.

In favorable cases, an isotropic foliation of the classical space of motions
$U$ can be lifted into a _Planck foliation_ of the quantum manifold $Y$, that
means a foliation such that the contact form defined on $Y$ vanishes on all
vectors tangent to the leaves. When a state vector $\Psi$ is constant on each
leaf of this Planck foliation, this vector state is said by the author to
_satisfy Planck’s condition_ (relative to the considered Planck foliation).
The _Planck space_ is the set of state vectors which satisfy Planck’s
condition. For a conservative system, using as isotropic foliation of $U$ the
foliation whose leaves are the integral curves of the Hamiltonian vector field
whose Hamiltonian is the energy, by writing explicitly Planck’s condition, the
author obtains the famous equality $E=h\nu$ which relates the energy $E$, the
Planck constant $h$ and the frequency $\nu$.

The author then assumes that a dynamical group $G$ acts on the classical
manifold of motions $U$, with a moment map, and that there exists an Abelian
normal subgroup $\widetilde{G}$ of $G$ whose symplectic cohomology is zero. He
uses as isotropic foliation of $U$ the foliation whose leaves are tangent to
the vector sub-bundle of $TU$ generated by Hamiltonian vector fields whose
Hamiltonians are the components of the moment map of the action of
$\widetilde{G}$. He lifts this foliation into a Planck foliation of $Y$,
writes explicitly the corresponding Planck’s condition, and determines the
associated Planck space. For an isolated system, $G$ will be either the
restricted Poincaré group or the Galilei group, according to whether the
system is relativistic or non-relativistic. The Abelian subgroup
$\widetilde{G}$ will be the group of space-time translations. The author
explains how Planck’s condition leads to the quantum wave equations. For a
non-relativistic material point, the author obtains by this means the
_Schrödinger equation_ , and for a relativistic material point, the _Klein-
Gordon equation_. For a non-relativistic particle with spin $1/2$, using a
$\mathbb{C}^{2}$-valued variable as state vector, the author proves that both
its components satisfy a Schrödinger equation; it is the description of
electrons proposed by Pauli. For a relativistic particle of spin $1/2$, he
obtains _Dirac’s equation_ and proves that rather than the Poincaré group
itself, it is a quantizable extension of this group whih acts on the set of
solutions of this equation. For him, the non-quantizability of the Poincaré
group provides a natural explanation of this fact, well known to physicists.
The author applies the same procedure for a massless particle, first with spin
$1/2$, then with spin $1$.

Next the author considers an assembly of particles of a given type in
indeterminate number. He denotes by $U_{\Phi}$ its manifold of motions and by
$U$ the manifold of motions of the system made by a single particle of the
considered type. He calls $U_{\Phi}$ the _Fock’s manifold_ and explains that
as a set, it is the set of all finite subsets of $U$. As a manifold,
$U_{\Phi}$ is the sum of disjoint parts of different dimensions, each part
$U_{n}$ being the manifold of motions of the system made of a fixed number $n$
of particles, with $0\leq n<+\infty$. For $U_{0}$, the author takes a
singleton considered as a $0$-dimensional manifold. As seen in the previous
section, for each $n\geq 2$, the part $U_{n}$ of $U_{\Phi}$ has two non-
equivalent quantizations, one for each distinct character of the group of
permutations of a set of $n$ elements. The author explains that for some
physical considerations, the same character should be chosen for all possible
values of $n$, therefore he obtains two non-equivalent quantizations
$Y_{\Phi}$ of $U_{\Phi}$. On the pre-Hilbert space ${\mathcal{H}}(Y_{\Phi})$,
he defines _creation_ and _annihilation_ operators, which respectively
increase or decrease the number of particles by one unit. Two different cases
must be distinguished, depending on whether the particles are _fermions_ or
_bosons_.

At the end of Chapter V, the author discusses the notion of _average value_ of
an observable for a given state vector. Planck’s condition appears as a
sufficient condition so that the statistical and quantum mechanical averages
of an observable coincide. Finally the author uses the notion of _function of
positive type_ defined on a group, with values in $\mathbb{C}$, to enlarge the
definitions of a _quantum state_ and of _average value_ of an observable, to
include states defined by a _density operator_ encountered in quantum
chemistry.

##  2 Comments

The great originality of the book _Structure des systèmes dynamiques_ clearly
appears in the detailed presentation of its contents given above. In this
section, we will first try to identify the most innovative and promising ideas
which can be found in it. Then we will write a few words about the terminology
and the notations used by the author, which do not lack originality too.

### Remarkabe scientific concepts.

Innovative ideas presented in the first two chapters are mainly related to the
ways in which difficult subjects can be taught to beginner students. The first
really remarkable scientific concept which appears in this book is, in our
opinion, presented in Chapter III. The author begins with a short and rather
classical account of the general principles of classical mechanics. Then he
proves that on the evolution space of a dynamical system, there exists a
remarkable presymplectic form, which he calls the _Lagrange form_. Its kernel
determines the vector field whose integral curves are the _motions_ of the
system. This form projects onto the set of all possible motions, called the
_space of motions_ , and its projection is a _symplectic form_ , _i.e._ , a
closed, non degenerate $2$-form. The author chooses this property, called the
_Maxwell principle_ , as the founding principle of mechanics. For us, it is a
very important innovation: the traditionally used concepts, such as the
_configuration space_ , the _evolution space_ , the _phase space_ , go into
the background and leave the first place to the _space of motions_ and to the
_Lagrange form_ of which it is endowed. Thanks to this innovation, the author
will be able to describe, in the framework of classical mechanics, with the
dynamics of a particle with spin, though there is no Lagrangian for such a
system, and with assemblies of an indeterminate number of such particles.
Similarly, he will be able to describe systems made by massless relativistic
particles, though there is no evolution space for such systems.

The space of motions of a dynamical system is not very often considered by
modern authors, though it appeared as soon as 1808 in the works of Lagrange.
This very natural concept has a nice mathematical property: the space of
motions is always endowed with a smooth manifold structure. The _phase
portrait_ of a dynamical system, a frequently used closely related concept,
very often has a much more complicated structure. One may wonder why the
concept of space of motions is not used more by modern authors. Maybe it is
because for some dynamical systems, the space of motions is a _non-Hausdorff_
manifold. Another possible explanation is that some scientists are interested
in the thorough description of particular motions of a system, rather than by
the study of the set of all possible motions. By showing that many important
results can be deduced from the symmetries of the space of motions of a
system, the author proves that this reluctance is ungrounded. For example, in
Chapter III, section 12, he proves that the principle of equality of action
and reaction appears as a consequence of Galilean relativity and Maxwell’s
principle.

In classical (non-relativistic) mechanics, the cohomological interpretation of
the _mass_ of an isolated dynamical system is, in our opinion, another
innovative idea worth mentioning. A cohomology class appears indeed in the
mathematical expression of the action of the Galilei group on the space of
motions of such a system. It is an element of a one-dimensional vector space,
and the author has proven that it can be interpreted as the mass of the
system. In classical mechanics, it is legitimate to consider isolated systems
with a positive, null or negative mass. It will be done by the author in his
study of the behaviour of elementary systems with respect to time or space
reversals.

Another originality of this book is that it presents dynamical systems in the
framework of relativistic physics as well as in the framework of classical,
non-relativistic mechanics. This is made possible thanks to the use of the
concept of state of motions. In classical, non-relativistic mechanics, the
Kepler problem and the $n$-bodies problem are very good mathematical models
for systems of gravitationally interacting material points. In relativistic
physics, there is no such mathematical model for a system of interacting
electrically charged particles. These particles interact by means of the
electomagnetic field created by their motion. Once created, this
electromagnetic field evolves according to its own laws. Up to now, no
mathematical model is available to describe the motions of the particles with
the integral curves of a vector field defined on some hypothetic evolution
space, only depending on the positions and motions of the particles. The
author manages to get around this difficulty by using the space of motions of
the system and Maxwell’s principle.

In classical mechanics, the symmetry group of the space of motions of an
isolated dynamical system is the Galilei group. In relativistic physics, this
group is the Poincaré group. This change of symmetry group has important
consequences clearly described by the author. The status of _mass_ is in
relativistic physics very different from its status in classical mechanics,
due to the fact that the symplectic cohomology of the Poincaré group is
trivial. In relativistic physics, there is no more a _barycentric
decomposition_ of the state of motions of an isolated system, as in classical
mechanics, because the Poincaré group has no privileged normal Abelian
subgroup, as the Galilean group has.

The works of the German mathematician Emmy Noether [9] told us that _first
integrals_ of a Lagrangian or Hamiltonian system very often are linked with
symmetries of the equations of motion. Most textbooks in classical mechanics
published before the author’s book only indicate how a real valued first
integral is determined by each one-parameter symmetry group of these
equations. We believe that Jean-Marie Souriau is, with Stephen Smale [10],
among the first scientists who considered the geometric structure of the full
set of these first integrals: they are the components of the _moment map_ of
the symmetry group’s action, defined on the evolution space and valued in the
dual vector space of the Lie algebra of this group.

An _elementary system_ is a relativistic dynamical system whose space of
motions is an homogeneous space of the restricted Poincaré group. This purely
mathematical definition, due to the author, seems to us very important because
the classification of elementary systems reveals many properties of physicits’
_elementary particles_ , especially their _mass_ , their _spin_ and, for
massless particles, their _helicity_ (whose possible values, $1$ or $-1$,
correspond to the two different circular polarizations of photons). In the
non-relativistic approximation, elementary systems are models of elementary
particles in the framework of classical mechanics. By this means, Geometric
Optics appears, when the particle’s spin is negligible, as a part of classical
mechanics. The author’s works so appear in continuity with those of Hamilton
[11], who introduced the _characteristic function_ first in Optics before
using it in mechanics. Interested readers will find in the long Introduction
of the book [12] a very nice discussion of the symplectic aspects of geometric
Optics and Electromagnetism.

Chapter IV also contains several worth mentioning innovations. For the author,
a _statistical state_ of a dynamical system is a probability measure on the
space of motions of the system. This definition is more natural than that
generally used, for example by George Mackey [13] who, instead of the state of
motions, uses the phase space at a given time. However, the most important
innovation contained in this chapter seems to us the generalization of the
notion of a Gibbs state, in which the energy is replaced by the moment map or
the action of a symmetry group. This very natural generalization (the energy
being the moment map of the action of the one-dimensional group of time
translations) involves remarkable new features when the symmetry group is not
Abelian. This group acts on the dual space of its Lie algebra by an affine
action involving a symplectic cocycle, with the coadjoint action as linear
part, for which the moment map is equivariant. By using generalized Gibbs
states, the author develops a kind of _Lie groups thermodynamics_ , which
seems to us very interesting, as well from the mathematician’s viewpoint as
for possible applications in physics. In this theory, the generalized
temperature and the generalized quantity of heat are, respectively, elements
of the Lie algebra of the symmetry group and of its dual vector space. The
author proves that an open convex subset of the Lie algebra is endowed with a
Riemannian metric which plays an important part in Statistics and Information
theory.

The seeds of Geometric Quantization, presented in Chapter V, can be found in
George Mackey’s small book [13], published in 1963. Jean-Marie Souriau [1] and
Bertram Kostant [2], its main creators, independently proposed two slightly
different, but equivalent versions of this theory. On the basis made by a
symplectic manifold, Kostant defines a bundle whose fibres are complex lines,
endowed with a connection whose curvature is the symplectic form of its basis.
Similarly, Souriau defines a bundle whose fibres are circles, endowed with a
contact form $\omega$, whose Reeb vector field is tangent to the fibres.
Moreover, the exterior derivative of $\omega$ projects onto the symplectic
form of the basis. Souriau’s circle bundle is the _principal bundle_ which is
associated to Kostant’s complex line bundle, and the connection form of
Kostant’s line bundle is the contact form $\omega$ of Souriau. While Souriau
quantizes the _manifold of motions_ of a classical dynamical system, Kostant
quantizes its _phase space_ at a given time. However, this difference in their
choice of the quantized symplectic manifold may not be as important as it
seems, because Souriau often uses the local symplectomorphism which exists
between the space of motions and the phase space, especially when he derives
the quantized wave equations.

Many remarkable results are presented in Chapter V. For instance, the author
proves that a quantizable system made by indistinguishable particles has
exactly two non-equivalent quantizations, both experimentally observed: they
are the Fermi-Dirac and Bose-Einstein quantizations well known by physicists.
He proves too that wave functions of quantum mechanics can be described with
functions defined on the quantum manifold, and that the usual spatio-temporal
description is obtained by means of a Fourier transform. The quantization of
the space of motions of an assembly of indistinguishable particles leads to
the _second quantization formalism_ : quantum vacuum, creation and
annihilation operators. The author also shows that the geometric obstructions
encountered when the action of a symmetry group on the classical space of
motions of a system is lifted to the quantum manifold, explains some facts
well known by physicists: the lift of the Abelian group of translations is the
non-Abelian _Weyl group_ ; elements of the Galilean group can be separately
lifted, but this full group does not act on the quantum manifold (fact due to
a cohomological obstruction discovered by Valentine Bargmann [5]); the
Poincaré group can be lifted for a particle of integer spin, but not for a
particle of half-integer spin, in which case it is a two sheets covering space
of the Poincaré group which can be lifted, which leads to the use of Dirac’s
spinors.

Finally, we must quote the many conjectural ideas presented by the author in
the last section of his book, for example about the behaviour of elementary
systems with respect to space or time reversals, and about a generalization of
the notion of quantum state.

### Language and notations of the author

The author’s language, although slightly different from the one generally used
in differential geometry, is perfectly logical and understandable. One of the
very few terms which could cause misunderstandings is that of _embedding_ (in
French, _plongement_). For the author, an embedding is a smooth map defined on
a smooth manifold $V$, or on an open subset of $V$, with values in another
smooth manifold $V^{\prime}$, which is injective (the author uses the term
_regular_ for _injective_) and whose rank is everywhere equal to the dimension
of $V$. Most geometers call such a map _an injective immersion_ , and use the
term _embedding_ for injective immersions which, in addition, are
homeomorphisms of their domain of definition onto their image, endowed with
the topology induced by that of $V^{\prime}$. It seems to us that the choice
made by the author is very reasonable because injective immersions are much
more frequently encountered than embeddings. For example, orbits of a Lie
group action, as well as leaves of a foliation, always are immersed in the
manifold in which they are contained, and much more rarely embedded. This
indication only concerns the original version of the book in French. In its
English translation [14], the translators use the usual term _injective
immersion_ and, in a footnote, indicate that for the author a _submanifold_ is
an _immersed submanifold_.

Another particularity of the author’s language which seems to us worth
mentioning, is about the notion of an _Euclidean vector space_ of finite
dimension. For the author, it is a real or complex finite-dimensional vector
space $E$ endowed with a symmetric, bilinear form $g$ satisfying the following
conditions.

  * •

If $E$ is a real vector space, $g$ is assumed to be non degenerate, which
means that for any $x\in E$, $x\neq 0$, there exits $y\in E$ such that
$g(x,y)\neq 0$. The author does not impose the condition $g$ positive. He
still uses the term _Euclidean vector space_ , not the term _pseudo-Euclidean
vector space_ when $g$ is definite without being positive, as done by most
mathematicians.

  * •

If $E$ is a complex $n$-dimensional vector space, $g$ is a symmetric, bilinear
form _for the structure of $2n$-dimensional real vector space of_ $E$
underlying its structure of $n$-dimensional complex vector space. The form $g$
is assumed to be non degenerate and to satisfy, for all pair $(x,y)$ of
elements in $E$, $g(ix,iy)=g(x,y)$, where $i=\sqrt{-1}$.

This convention seems to us very logical and useful, because it unifies two
different concepts often taught separately: the concept of real Euclidean
finite-dimensional vector space and that of complex finite-dimensional
Hermitian vector space.

Some notations of the author seem to us disconcerting. For example, he calls
_variable_ any symbol, such as the letter $y$, and associates it to a map. He
denotes by $[\hbox{value}\ y](a)$ the value of the map associated to $y$ at
the point $a$ if its domain of definition. He uses different symbols for the
variable, the associated map, the point of its domain of definition and its
value at that point. So it is sometimes difficult to understand some
expressions he writes, because the reader must simultaneously have in mind the
meaning of many symbols.

Given a vector field $f$ defined on a smooth manifold $V$, the author calls
_derivation_ associated to $f$ the operation which, to any smooth map $A$
defined on $V$ and valued in a smooth manifold $V^{\prime}$, associates the
map whose value, at each $x\in V$, is the image of the vector $f(x)$ by the
linear map tangent to $A$ at $x$. He denotes derivations by symbols such as
$d$ or $\delta$, other than the symbol which denotes the vector field (here
the symbol which denotes the vector field is $f$). When the author considers
several vector fields, he uses symbols such as $\mathrm{d}$, $\mathrm{d}_{1}$,
$\delta$, $\delta_{1}$, etc to denote the corresponding derivations. Of
course, these conventions are perfectly logical and coherent. However, their
use may make the reading of some formulae rather difficult. Moreover, since
the symbol $\mathrm{d}$ is used for derivations associated to vector fields,
the author cannot use it for the exterior derivation of differential forms, as
done by most mathematicians. For the exterior derivation of differential
forms, the author uses the symbol $\nabla$, which may disconcert some readers.

Let us finally indicate a few pecularities of the notations used by the author
in exterior differential calculus. He does not use the symbol $\wedge$ for the
exterior product of differential forms, which makes some formulae heavy. He
denotes by $\eta(X)$ the interior product of a differential form $\eta$ by a
vector field $X$, while many other mathematicians denote it by
$\mathrm{i}(X)\eta$, $\mathrm{i}_{X}\eta$ or $\mathrm{i}_{X}(\eta)$. It is
probably for this reason that he denotes by $\eta(X_{1})(X_{2})\cdots(X_{p})$
the evaluation of the $p$-form $\eta$ on the vector fields
$X_{1},X_{2},\ldots,X_{p}$, instead of writing more simply
$\eta(X_{1},\ldots,X_{p})$. At first glance, this convention may seem simple
and convenient. However, it makes some formulae heavy, such as the Cartan
formula ${\mathcal{L}}(X)=\mathrm{i}(X)\mathrm{d}+\mathrm{d}\mathrm{i}(X)$,
which expresses the Lie derivative with respect to a vector field $X$ as the
anticommutator of the interior product by $X$ with the exterior derivative.

##  3 Novel research ideas in Jean-Marie Souriau’s footsteps

Every research book is a survey at a given time of the state of knowledge
obviously limited on the issue. Just as the author considered that the book by
Lagrange Mécanique analytique was unfinished, we believe that Structure des
systèmes dynamique is a very in-depth work opening new research paths.

##  4 Conclusion

We cannot fail to be impressed when reading this book by the extent and the
thoroughness of the author’s knowledge, as well in mathematics as in mechanics
or in physics, and by the originality and the depth of his thoughts. Jean-
Marie Souriau is the author of two other very remarkable books, _Géométrie et
Relativité_ [15] and _Calcul linéaire_ [7], which are very rich and original
too and deserve to be read, and read again. We believe that among the paths
for research he indicates in these books, many still are not yet fully
explored.

## References

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  * [2] B. Kostant, ”Quantization and Unitary Representations, part 1, Prequantization”, Lecture Notes in Mathematics 170 (1970), p. 87-208.
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