From tools in symplectic and Poisson geometry to Souriau's theories of statistical mechanics and thermodynamics

I present in this paper some tools in Symplectic and Poisson Geometry in view of their applications in Geometric mechanics and Mathematical Physics. After a short discussion of the Lagrangian and Hamiltonian formalisms, including the use of symmetry groups, and a presentation of the Tulczyjew's isomorphisms (which explain some aspects of the relations between these formalisms), I explain the concept of manifold of motions of a mechanical system and its use, due to J.-M. Souriau, in Statistical Mechanics and Thermodynamics. The generalization of the notion of thermodynamic equilibrium in which the one-dimensional group of time translations is replaced by a multi-dimensional, maybe non-commutative Lie group, is discussed and examples of applications in Physics are given.

1 Introduction 1.1 Contents of the paper, sources and further reading This paper presents tools in Symplectic and Poisson Geometry in view of their application in Geometric Mechanics and Mathematical Physics. The Lagrangian formalism and symmetries of Lagrangian systems are discussed in Sections 2 and 3, the Hamiltonian formalism and symmetries of Hamiltonian systems in Sections 4 and 5. Section 6 introduces the concepts of Gibbs state and of thermodynamic equilibrium of a mechanical system, and presents several examples. For a monoatomic classical ideal gas, eventually in a gravity field, or a monoatomic relativistic gas the Maxwell-Boltzmann and Maxwell-Jüttner probability distributions are derived. The Dulong and Petit law which governs the specific heat of solids is obtained. Finally Section 7 presents the generalization of the concept of Gibbs state, due to Jean-Marie Souriau, in which the group of time translations is replaced by a (multi-dimensional and eventually non-Abelian) Lie group.
Several books [1,2,12,15,18,19,30,31,45,63] discuss, much more fully than in the present paper, the contents of Sections 2 to 5. The interested reader is referred to these books for detailed proofs of results whose proofs are only briefly sketched here. The recent paper [38] contains detailed proofs of most results presented here in Sections 4 and 5.
The main sources used for Sections 6 and 7 are the book and papers by Jean-Marie Souriau [55,54,56,57,58] and the beautiful small book by G. W. Mackey [34].
The Euler-Poincaré equation, which is presented with Lagrangian symmetries at the end of Section 3, is not really related to symmetries of a Lagrangian system, since the Lie algebra which acts on the configuration space of the system is not a Lie algebra of symmetries of the Lagrangian. Moreover in its intrinsic form that equation uses the concept of Hamiltonian momentum map presented later, in Section 5. Since the Euler-Poincaré equation is not used in the following sections, the reader can skip the corresponding subsection at his or her first reading.

Notations
The notations used are more or less those generally used now in Differential Geometry. A smooth, time-dependent vector field on the smooth manifold M is a smooth map X : R × M → T M such that, for each t ∈ R and x ∈ M, X (t, x) ∈ T x M, the vector space tangent to M at X . When, for any x ∈ M, X (t, x) does not depend on t ∈ R, X is a smooth vector field in the usual sense, i.e., an element in A 1 (M). Of course a time-dependent vector field can be defined on an open subset of R × M instead than on the whole R × T M. It defines a differential equation said to be associated to X . The (full) flow of X is the map Ψ X , defined on an open subset of R×R×M, taking its values in M, such that for each t 0 ∈ R and x 0 ∈ M the parametrized curve t → Ψ X (t,t 0 , x 0 ) is the maximal integral curve of ( * ) satisfying Ψ(t 0 ,t 0 , x 0 ) = x 0 . When t 0 and t ∈ R are fixed, the map x 0 → Ψ X (t,t 0 , x 0 ) is a diffeomorphism, defined on an open subset of M (which may be empty) and taking its values in another open subset of M, denoted by Ψ X (t,t 0 ) . When X is in fact a vector field in the usual sense (not dependent on time), Ψ X (t,t 0 ) only depends on t − t 0 . Instead of the full flow of X we can use its reduced flow Φ X , defined on an open subset of R × M and taking its values in M, related to the full flow Ψ X by  The principles of Mechanics were stated by the great English mathematician Isaac Newton (1642Newton ( -1727 in his book Philosophia Naturalis Principia Mathematica published in 1687 [44]. On this basis, a little more than a century later, Joseph Louis Lagrange (1736Lagrange ( -1813 in his book Mécanique analytique [28] derived the equations (today known as the Euler-Lagrange equations) which govern the motion of a mechanical system made of any number of material points or rigid material bodies interacting between them by very general forces, and eventually submitted to external forces.
In modern mathematical language, these equations are written on the configuration space and on the space of kinematic states of the considered mechanical system. The configuration space is a smooth n-dimensional manifold N whose elements are all the possible configurations of the system (a configuration being the position in space of all parts of the system). The space of kinematic states is the tangent bundle T N to the configuration space, which is 2n-dimensional. Each element of the space of kinematic states is a vector tangent to the configuration space at one of its elements, i.e. at a configuration of the mechanical system, which describes the velocity at which this configuration changes with time. In local coordinates a configuration of the system is determined by the n coordinates x 1 , . . ., x n of a point in N, and a kinematic state by the 2n coordinates x 1 , . . . , x n , v 1 , . . . v n of a vector tangent to N at some element in N.

The Euler-Lagrange equations
When the mechanical system is conservative, the Euler-Lagrange equations involve a single real valued function L called the Lagrangian of the system, defined on the product of the real line R (spanned by the variable t representing the time) with the manifold T N of kinematic states of the system. In local coordinates, the Lagrangian L is expressed as a function of the 2n + 1 variables, t, x 1 , . . . , x n , v 1 , . . ., v n and the Euler-Lagrange equations have the remarkably simple form d dt where x(t) stands for x 1 (t), . . ., x n (t) and v(t) for v 1 (t), . . ., v n (t) with, of course,

Hamilton's principle of stationary action
The great Irish mathematician William Rowan Hamilton (1805-1865) observed [16,17] that the Euler-Lagrange equations can be obtained by applying the standard techniques of Calculus of Variations, due to Leonhard Euler (1707-1783) and Joseph Louis Lagrange, to the action integral 1 where γ : [t 0 ,t 1 ] → N is a smooth curve in N parametrized by the time t. These equations express the fact that the action integral I L (γ) is stationary with respect to any smooth infinitesimal variation of γ with fixed end-points t 0 , γ(t 0 ) and t 1 , γ(t 1 ) . This fact is today called Hamilton's principle of stationary action. The reader interested in Calculus of Variations and its applications in Mechanics and Physics is referred to the books [8,10,29].

The Euler-Cartan theorem
The Lagrangian formalism is the use of Hamilton's principle of stationary action for the derivation of the equations of motion of a system. It is widely used in Mathematical Physics, often with more general Lagrangians involving more than one independent variable and higher order partial derivatives of dependent variables. For simplicity I will consider here only the Lagrangians of (maybe time-dependent) conservative mechanical systems.
An intrinsic geometric expression of the Euler-Lagrange equations, wich does not use local coordinates, was obtained by the great French mathematicianÉlie Cartan . Let us introduce the concepts used by the statement of this theorem.

Definitions.
Let N be the configuration space of a mechanical system and let its tangent bundle T N be the space of kinematic states of that system. We assume that the evolution with time of the state of the system is governed by the Euler-Lagrange equations for a smooth, maybe time-dependent Lagrangian L : R × T N → R.
1. The cotangent bundle T * N is called the phase space of the system.
is called the the energy function associated to L.
where θ N is the Liouville 1-form on T * N, is called the Euler-Poincaré 1-form.
The interested reader will find the proof of that theorem in [35], (theorem 2.2, chapter IV, page 262) or, for hyper-regular Lagrangians (an additional assumption which in fact, is not necessary) in [59], chapter IV, theorem 2.1 page 167.

Remark.
In his book [55], Jean-Marie Souriau uses a slightly different terminology: for him the odd-dimensional space R × T N is the evolution space of the system, and the exact 2-form d ϖ L on that space is the Lagrange form. He defines that 2-form in a setting more general than that of the Lagrangian formalism.

Lagrangian symmetries 3.1 Assuumptions and notations
In this section N is the configuration space of a conservative Lagrangian mechanical system with a smooth, maybe time dependent Lagrangian L : R × T N → R. Let ϖ L be the Poincaré-Cartan 1-form on the evolution space R × T N.
Several kinds of symmetries can be defined for such a system. Very often, they are special cases of infinitesimal symmetries of the Poincaré-Cartan form, which play an important part in the famous Noether theorem.
L(Z) denoting the Lie derivative of differential forms with respect to Z.
Let us assume that the Lagrangian L does not depend on the time t ∈ R, i.e. is a smooth function on T N. The vector field on R × T N denoted by ∂ ∂t , whose projection on R is equal to 1 and whose projection on T N is 0, is an infinitesimal symmetry of ϖ L .

2.
Let X be a smooth vector field on N and X be its canonical lift to the tangent bundle T N. We still assume that L does not depend on the time t. Moreover we assume that X is an infinitesimal symmetry of the Lagrangian L, i.e. that L(X)L = 0. Considered as a vector field on R × T N whose projection on the factor R is 0, X is an infinitesimal symmetry of ϖ L .

The
Since Z is an infinitesimal symmetry of ϖ L , Using the well known formula relating the Lie derivative, the interior product and the exterior derivative

The Lagrangian momentum map
The Lie bracket of two infinitesimal symmetries of ϖ L is too an infinitesimal symmetry of ϖ L . Let us therefore assume that there exists a finite-dimensional Lie algebra of vector fields on R × T N whose elements are infinitesimal symmetries of ϖ L .
be a Lie algebras homomorphism of a finitedimensional real Lie algebra G into the Lie algebra of smooth vector fields on R × T N such that, for each X ∈ G, ψ(X ) is an infinitesimal symmetry of ϖ L . The Lie algebras homomorphism ψ is said to be a Lie algebra action on R×T N by infinitesimal symmetries of ϖ L . The map K L : R × T N → G * , which takes its values in the dual G * of the Lie algebra G, defined by is called the Lagrangian momentum of the Lie algebra action ψ.

The Euler-Poincaré equation
In a short Note [47] published in 1901, the great french mathematician Henri Poincaré (1854Poincaré ( -1912 proposed a new formulation of the equations of Mechanics. Let N be the configuration manifold of a conservative Lagrangian system, with a smooth Lagrangian L : T N → R which does not depend explicitly on time. Poincaré assumes that there exists an homomorphism ψ of a finite-dimensional real Lie algebra G into the Lie algebra A 1 (N) of smooth vector fields on N, such that for each x ∈ N, the values at x of the vetor fields ψ(X ), when X varies in G, completely fill the tangent space T x N. The action ψ is then said to be locally transitive.
Of course these assumptions imply dim G ≥ dim N.
Under these assumptions, Henri Poincaré proved that the equations of motion of the Lagrangian system could be written on N × G or on N × G * , where G * is the dual of the Lie algebra G, instead of on the tangent bundle T N. When dim G = dim N (which can occur only when the tangent bundle T N is trivial) the obtained equation, called the Euler-Poincaré equation, is perfectly equivalent to the Euler-Lagrange equations and may, in certain cases, be easier to use. But when dim G > dim N, the system made by the Euler-Poincaré equation is underdetermined.
When dim G > dim N the smooth curve V in G is not uniquely determined by the smooth curve γ in N. However, instead of writing the second-order Euler-Lagrange differential equations on T N satisfied by γ when this curve is a possible motion of the Lagrangian system, Poincaré derives a first order differential equation for the curve V and proves that it is satisfied, together with Equation ( * ), if and only if γ is a possible motion of the Lagrangian system. Let ϕ : N × G → T N and L : N × G → R be the maps We denote by d 1 L : N × G → T * N and by d 2 L : N × G → G * the partial differentials of L : N ×G → R with respect to its first variable x ∈ N and with respect to its second variable X ∈ G. The map ϕ : N × G → T N is a surjective vector bundles morphism of the trivial vector bundle N × G into the tangent bundle T N. Its transpose ϕ T : T * N → N × G * is therefore an injective vector bundles morphism, which can be written where π N : T * N → N is the canonical projection of the cotangent bundle and J : T * N → G * is a smooth map whose restriction to each fibre T * x N of the cotangent bundle is linear, and is the transpose of the map X → ϕ(x, X ) = ψ(X )(x).
The curve γ is a possible motion of the Lagrangian system if and only if V satisfies the equation The interested reader will find a proof of that theorem in local coordinates in the original Note by Poincaré [47]. More intrinsic proofs can be found in [37,38]. Another proof is possible, in which that theorem is deduced from the Euler-Cartan theorem 2.4.2.

The Hamiltonian formalism
The Lagrangian formalism can be applied to any smooth Lagrangian. Its application yields second order differential equations on R × T N (in local coordinates, the Euler-Lagrange equations) which in general are not solved with respect to the second order derivatives of the unknown functions with respect to time. The classical existence and unicity theorems for the solutions of differential equations (such as the Cauchy-Lipschitz theorem) therefore cannot be applied to these equations.
Under the additional assumption that the Lagrangian is hyper-regular, a very clever change of variables discovered by William Rowan Hamilton 2 [16,17] allows a new formulation of these equations in the framework of symplectic geometry. The Hamiltonian formalism discussed below is the use of these new equations. It was later generalized independently of the Lagrangian formalism.

Assumptions made in this section
We consider in this section a smooth, maybe time-dependent Lagrangian L : R ×T N → R, which is such that the Legendre map (2.4.1) L L : R × T N → T * N satisfies the following property: for each fixed value of the time t ∈ R, the map v → L L (t, v) is a smooth diffeomorphism of the tangent bundle T N onto the cotangent bundle T * N. An equivalent assumption is the following: the map (id R , The Lagrangian L is then said to be hyper-regular. The equations of motion can be written on R × T * N instead of R × T N.

Remarks.
A symplectic form ω on a manifold M is a presymplectic form which, moreover, is non-degenerate, i.e. such that for each x ∈ M and each non-zero vector v ∈ T x M, there exists another vector w ∈ T x M such that ω(x)(v, w) = 0. Or in other words, a presymplectic form ω whose kernel is the set of null vectors.
The kernel of a presymplectic form ω on a smooth manifold M is a vector sub-bundle of T M if and only if for each x ∈ M, the vector subspace T x M of vectors v ∈ T x M which satisfy i(v)ω = 0 is of a fixed dimension, the same for all points x ∈ M. A presymplectic form which satisfies that condition is said to be of constant rank. We now assume in addition that this foliation is simple, i.e. such that the set of leaves of F ω , denoted by M/ ker ω, has a smooth manifold structure for which the canonical projection p : M → M/ ker ω (which associates to each point x ∈ M the leaf which contains x) is a smooth submersion. There exists on M/ ker ω a unique symplectic form ω r such that ω = p * ω r .
Proof. Since dω = 0, the fact that ker ω is completely integrable is an immediate consequence of the Frobenius' theorem ( [59], chapter III, theorem 5.1 page 132). The existence and unicity of a symplectic form ω r on M/ ker ω such that ω = p * ω r results from the fact that M/ ker ω is built by quotienting M by the kernel of ω.

Presymplectic manifolds in Mechanics
Their exterior differentials d ϖ L and d ϖ H L both are presymplectic 2-forms on the odddimensional manifolds R × T N and R × T * N, respectively. At any point of these manifolds, the kernels of these closed 2 forms are one-dimensional. They therefore (4.2.3) determine foliations into smooth curves of these manifolds. The Euler-Cartan theorem (2.4.2) shows that each of these curves is a possible motion of the system, described either in the Lagrangian formalism, or in the Hamiltonian formalism, respectively.
The set of all possible motions of the system, called by Jean-Marie Souriau the manifold of motions of the system, is described by the quotient (R × T N)/ ker d ϖ L in the Lagrangian formalism, and by the quotient (R × T * N)/ ker d ϖ H L in the Hamiltonian formalism. Both are (maybe non-Hausdorff) symplectic manifolds, the projections on these quotient manifolds of the presymplectic forms d ϖ L and d ϖ H L both being symplectic forms. Of course the diffeomorphism (id R , L L ) : R × T N → R × T * N projects onto a symplectomorphism between the Lagrangian and Hamiltonian descriptions of the manifold of motions of the system.

The Hamilton equation
4.3.1 Proposition. Let N be the configuration manifold of a Lagrangian system whose Lagrangian L : R × T N → R, maybe time-dependent, is smooth and hyper-regular, and H L : R × T * N → R be the associated Hamiltonian (4.1.2). Let ϕ : [t 0 ,t 1 ] → N be a smooth curve parametrized by the time t ∈ [t 0 ,t 1 ], and let ψ : The parametrized curve t → γ(t) is a motion of the system if and only if the parametrized curve t → ψ(t) satisfies the equatin, called the Hamilton equation, Proof. These results directly follow from the Euler-Cartan theorem (2.4.2).

The Tulczyjew isomorphisms
Around 1974, W.M. Tulczyjew [61,62] discovered 3 two remarkable vector bundles isomorphisms α N : T T * N → T * T N and β N : The first one α N is an isomorphism of the bundle (T T * N, T π N , T N) onto the bundle (T * T N, π T N , T N), while the second β N is an isomorphism of the bundle (T T * N, τ T * N , T * N) onto the bundle (T * T * N, π T * N , T * N). The diagram below is commutative.
y y s s s s s s s s s s s N Since they are the total spaces of cotangent bundles, the manifolds T * T N and T * T * N are endowed with the Liouville 1-forms θ T N and θ T * N , and with the canonical symplectic forms dθ T N and dθ T * N , respectively. Using the isomorphisms α N and β N , we can therefore define on T T * N two 1-forms α * N θ T N and β * N θ T * N , and two symplectic 2-forms α * N (dθ T N ) and β * N (dθ T * N ). The very remarkable property of the isomorphisms α N and β N is that the two symplectic forms so obtained on T T * N are equal: The 1-forms α * N θ T N and β * N θ T * N are not equal, their difference is the differential of a smooth function.

Lagrangian submanifolds
In view of applications to implicit Hamiltonian systems, let us recall here that a Lagrangian submanifold of a symplectic manifold (M, ω) is a submanifold N whose dimension is half the dimension of M, on which the form induced by the symplectic form ω is 0.
Let L : T N → R and H : T * N → R be two smooth real valued functions, defined on T N and on T * N, respectively. The graphs dL(T N) and dH(T * N) of their differentials are Lagrangian submanifolds of the symplectic manifolds (T * T N, dθ T N ) and (T * T * N, dθ T * N ). Their pull-backs α −1 N dL(T N) and β −1 N dH(T * N) by the symplectomorphisms α N and β N are therefore two Lagrangian submanifolds of the manifold T T * N endowed with the symplectic form α * N (dθ T N ), which is equal to the symplectic form β * N (dθ T * N ). The following theorem enlightens some aspects of the relationships between the Hamiltonian and the Lagrangian formalisms.

) holds if and only if the Lagrangian L is hyper-regular and such that
where L L : T N → T * N is the Legendre map and E L : T N → R the energy associated to the Lagrangian L.
The interested reader will find the proof of that theorem in the works of W. Tulczyjew ([61, 62]).
When L is not hyper-regular, α −1 N dL(T N) still is a Lagrangian submanifold of the symplectic manifold T T * N, α * N (dθ T N ) , but it is no more the graph of a smooth vector field X H defined on T * N. Tulczyjew proposes to consider this Lagrangian submanifold as an implicit Hamilton equation on T * N.
These results can be extended to Lagrangians and Hamiltonians which may depend on time.

The Hamiltonian formalism on symplectic and Poisson manifolds 4.4.1 The Hamilton formalism on symplectic manifolds
In pure mathematics as well as in applications of mathematics to Mechanics and Physics, symplectic manifolds other than cotangent bundles are encountered. A theorem due to the french mathematician Gaston Darboux (1842-1917) asserts that any symplectic manifold (M, ω) is of even dimension 2n and is locally isomorphic to the cotangent bundle to a ndimensional manifold: in a neighbourhood of each of its point there exist local coordinates (x 1 , . . ., x n , p 1 , . . ., p n ), called Darboux coordinates with which the symplectic form ω is expressed exactly as the canonical symplectic form of a cotangent bundle: Let (M, ω) be a symplectic manifold and H : R × M → R a smooth function, said to be a time-dependent Hamiltonian. It determines a time-dependent Hamiltonian vector field H t : M → R being the function H in which the variable t is considered as a parameter with respect to which no differentiation is made. The Hamilton equation determined by H is the differential equation The Hamiltonian formalism can therefore be applied to any smooth, maybe time dependent Hamiltonian on M, even when there is no associated Lagrangian. The Hamiltonian formalism is not limited to symplectic manifolds: it can be applied, for example, to Poisson manifolds [32], contact manifolds and Jacobi manifolds [33]. For simplicity I will consider only Poisson manifolds. Readers interested in Jacobi manifolds and their generalizations are referred to the papers by A. Lichnerowicz quoted above and to the very important paper by A. Kirillov [23].

Definition.
A Poisson manifold is a smooth manifold P whose algebra of smooth functions C ∞ (P, R) is endowed with a bilinear composition law, called the Poisson bracket, which associates to any pair ( f , g) of smooth functions on P another smooth function denoted by { f , g}, that composition satisfying the three properties

Examples.
1. On the vector space of smooth functions defined on a symplectic manifold (M, ω), there exists a composition law, called the Poisson bracket, which satisfies the properties stated in 4.4.2. Let us recall briefly its definition. The symplectic form ω allows us to associate, to any smooth function f ∈ C ∞ (M, R), a smooth vector field X f ∈ A 1 (M, R), called the Hamiltonian vector field associated to f , defined by Any symplectic manifold is therefore a Poisson manifold.
The Poisson bracket of smooth functions defined on a symplectic manifold (when that symplectic manifold is a cotangent bundle) was discovered by Siméon Denis Poisson (1781-1840) [48].
2. Let G be a finite-dimensional real Lie algebra, and let G * be its dual vector space.
For each smooth function f ∈ C ∞ (G * , R) and each ζ ∈ G * , the differential d f (ζ ) is a linear form on G * , in other words an element of the dual vector space of G * . Identifying with G the dual vector space of G * , we can therefore consider d f (ζ ) as an element in G. With this identification, we can define the Poisson bracket of two smooth functions f and the bracket in the right hand side being the bracket in the Lie algebra G. The Poisson bracket of functions in C ∞ (G * , R) so defined satifies the properties stated in 4.4.2. The dual vector space of any finite-dimensional real Lie algebra is therefore endowed with a Poisson structure, called its canonical Lie-Poisson structure or its Kirillov-Kostant-Souriau Poisson structure. Discovered by Sophus Lie, this structure was indeed rediscovered independently by Alexander Kirillov, Bertram Kostant and Jean-Marie Souriau.
The canonical Lie-Poisson bracket of two smooth functions f and g ∈ C ∞ (G * , R) can be modified by means of the symplectic cocycle Θ, by setting This bracket still satifies the properties stated in 4.4.2, therefore defines on G * a Poisson structure called its canonical Lie-Poisson structure modified by Θ.

Properties of Poisson manifolds
The interested reader will find the proofs of the properties recalled here in [63], [31], [30] or [45]. 1. On a Poisson manifold P, the Poisson bracket { f , g} of two smooth functions f and g can be expressed by means of a smooth field of bivectors Λ: called the Poisson bivector field of P. The considered Poisson manifold is often denoted by (P, Λ). The Poisson bivector field Λ identically satisfies in the left hand side being the Schouten-Nijenhuis bracket. That bivector field determines a vector bundle morphism Λ ♯ : T * P → T P, defined by where η and ζ ∈ T * P are two covectors attached to the same point in P.
Readers interested in the Schouten-Nijenhuis bracket will find thorough presentations of its properties in [25] or [36]. 2. Let (P, Λ) be a Poisson manifold. A (maybe time-dependent) vector field on P can be associated to each (maybe time-dependent) smooth function H : R × P → R. It is called the Hamiltonian vector field associated to the Hamiltonian H, and denoted by X H . Its expression is ∂t dt is the differential of the function deduced from H by considering t as a parameter with respect to which no differentiation is made.

Definition.
A smooth vector field X on a presymplectic (resp. symplectic, resp.
Not any smooth function on a presymplectic manifold can be a Hamiltonian.
where {H, g} = Λ(dH, dg) denotes the Poisson bracket of the functions H and g.
On a symplectic or a Poisson manifold, any smooth function can be a Hamiltonian.
The proof of this result, which is easy, can be found in any book on symplectic and Poisson geoemetry, for example [31], [30] or [45].

Lie algebras and Lie groups actions
Definitions. An action on the left (resp. an action on the right) of a Lie group G on a smooth manifold M is a smooth map Φ : G×M → M (resp. a smooth map Ψ : M ×G → M) such that • Φ e = id M (resp. Ψ e = id M ), e being the neutral element of G, An action of a Lie algebra G on a smooth manifold M is a Lie algebras morphism of G into the Lie algebra A 1 (M) of smooth vector fields on M, i.e. a linear map ψ : G → A 1 (M) which associates to each X ∈ G a smooth vector field ψ(X ) on M such that for each pair

5.2.2
Proposition. An action Ψ, either on the left or on the right, of a Lie group G on a smooth manifold M, automatically determines an action ψ of its Lie algebra G on that manifold, which associates to each X ∈ G the vector field ψ(X ) on M, often denoted by X M and called the fundamental vector field on M associated to X . It is defined by with the following convention: ψ is a Lie algebras homomorphism when we take for Lie algebra G of the Lie group G the Lie algebra or right invariant vector fields on G if Ψ is an action on the left, and the Lie algebra of left invariant vector fields on G if Ψ is an action on the right.
Proof. If Ψ is an action of G on M on the left (respectively, on the right), the vector field on G which is right invariant (respectively, left invariant) and whose value at e is X , and the associated fundamental vector field X M on M, are compatible by the map g → Ψ g (x). Therefore the map ψ : G → A 1 (M) is a Lie algebras homomorphism, if we take for definition of the bracket on G the bracket of right invariant (respectively, left invariant) vector fields on G.

Definitions.
When M is a presymplectic (or a symplectic, or a Poisson) manifold, an action Ψ of a Lie group G (respectively, an action ψ of a Lie algebra G) on the manifold M is called a presymplectic (or a symplectic, or a Poisson) action if for each g ∈ G, Ψ g is a presymplectic, or a symplectic, or a Poisson diffeomorphism of M (respectively, if for each X ∈ G, ψ(X ) is a presymplectic, or a symplectic, or a Poisson vector field on M.

5.2.4
Definitions. An action ψ of a Lie algeba G on a presymplectic or symplectic manifold (M, ω), or on a Poisson manifold (M, Λ), is said to be Hamiltonian if for each X ∈ G, the vector field ψ(X ) on M is Hamiltonian. An action Ψ (either on the left or on the right) of a Lie group G on a presymplectic or symplectic manifold (M, ω), or on a Poisson manifold (M, Λ), is said to be Hamiltonian if that action is presymplectic, or symplectic, or Poisson (according to the structure of M), and if in addition the associated action of the Lie algebra G of G is Hamiltonian.

Remark.
A Hamiltonian action of a Lie group, or of a Lie algebra, on a presymplectic, symplectic or Poisson manifold, is automatically a presymplectic, symplectic or Poisson action. This result immediately follows from 5.
The map J is called a momentum map for the Lie algebra action ψ. When ψ is the action of the Lie algebra G of a Lie group G associated to a Hamiltonian action Ψ of a Lie group G, J is called a momentum map for the Hamiltonian Lie group action Ψ.
The proof of that result, which is easy, can be found for example in [31], [30] or [45].

Remark.
The momentum map J is not unique: • when (M, ω) is a connected symplectic manifold, J is determined up to addition of an arbitrary constant element in G * ; • when (M, Λ) is a connected Poisson manifold, the momentum map J is determined up to addition of an arbitrary G * -valued smooth map which, coupled with any X ∈ G, yields a Casimir of the Poisson algebra of (M, Λ), i.e. a smooth function on M whose Poisson bracket with any other smooth function on that manifold is the function identically equal to 0. Proof. The function f is constant on each integral curve of X g if and only if i(X g )d f = 0, since each integral curve of X g is connected. We can use the Poisson bracket, even when M is a presymplectic manifold, since the Poisson bracket of two Hamiltonians on a presymplectic manifold still can be defined. So we can write

Corollary
The action A can be written, with g ∈ G and ξ ∈ G * , if Φ is an action on the right.
Proof. Let us assume that Φ is an action on the left. The fundamental vector field X M associated to each X ∈ G is Hamiltonian, with the function J X : M → R, given by Hamiltonian. An easy calculation shows that this vector field is the fundamental vector field associated to Ad g −1 (X ) ∈ G.
The function , X is therefore a Hamiltonian for that vector field. These two functions defined on the connected manifold M, which both are admissible Hamiltonians for the same Hamiltonian vector field, differ only by a constant (which may depend on g ∈ G). We can set, for any g ∈ G, and check that the map A : G × G * → G * defined in the statement is indeed an action for which J is equivariant.
A similar proof, with some changes of signs, holds when Φ is an action on the right.
More precisely θ is a symplectic cocycle. It means that its differential T e θ : T e G ≡ G → G * at the neutral element e ∈ G can be considered as a skew-symmetric bilinear form on G: The skew-symmetric bilinear form Θ is a symplectic cocycle of the Lie algebra G. It means that it is skew-symmetric and satisfies, for all X , Y and Z ∈ G, Proof. These properties easily follow from the fact that when Φ is an action on the left, for g and h ∈ G, Φ g • Φ h = Φ gh (and a similar equality when Φ is an action on the right).
(G being identified with its bidual G * * ), determines a Poisson structure on G * , and the momentum map J : M → G * is a Poisson map, M being endowed with the Poisson structure associated to its symplectic structure.
Proof. The fact that the bracket ( f , g) → { f , g} Θ on C ∞ (G * , R) is a Poisson bracket was already indicated in 4.4.3. It can be verified by easy calculations. The fact that J is a Poisson map can be proven by first looking at linear functions on G * , i.e. elements in G. The reader will find a detailed proof in [38].

Remark.
When the momentum map J is replaced by another momentum map J 1 = J + µ, where µ ∈ G * is a constant, the symplectic Lie group cocycle θ and the symplectic Lie algebra cocycle Θ are replaced by θ 1 and Θ 1 , respectively, given by These formulae show that θ 1 − θ and Θ 1 − Θ are symplectic coboudaries of the Lie group G and the Lie algebra G. In other words, the cohomology classes of the cocycles θ and Θ only depend on the Hamiltonian action Φ of G on the symplectic manifold (M, ω).

Symmetries of the phase space
Hamiltonian Symmetries are often used for the search of solutions of the equations of motion of mechanical systems. The symmetries considered are those of the phase space of the mechanical system. This space is very often a symplectic manifold, either the cotangent bundle to the configuration space with its canonical symplectic structure, or a more general symplectic manifold. Sometimes, after some simplifications, the phase space is a Poisson manifold.
The Marsden-Weinstein reduction procedure [39,43] or one of its generalizations [45] is the method most often used to facilitate the determination of solutions of the equations of motion. In a first step, a possible value of the momentum map is chosen and the subset of the phase space on which the momentum map takes this value is determined. In a second step, that subset (when it is a smooth manifold) is quotiented by its isotropic foliation. The quotient manifold is a symplectic manifold of a dimension smaller than that of the original phase space, and one has an easier to solve Hamiltonian system on that reduced phase space.
When Hamiltonian symmetries are used for the reduction of the dimension of the phase space of a mechanical system, the symplectic cocycle of the Lie group of symmetries action, or of the Lie algebra of symmetries action, is almost always the zero cocycle.
For example, if the goup of symmetries is the canonical lift to the cotangent bundle of a group of symmetries of the configuration space, not only the canonical symplectic form, but the Liouville 1-form of the cotangent bundle itself remains invariant under the action of the symmetry group, and this fact implies that the symplectic cohomology class of the action is zero.

Symmetries of the space of motions
A completely different way of using symmetries was initiated by Jean-Marie Souriau, who proposed to consider the symmetries of the manifold of motions of the mechanical system. He observed that the Lagrangian and Hamiltonian formalisms, in their usual formulations, involve the choice of a particular reference frame, in which the motion is described. This choice destroys a part of the natural symmetries of the system. For example, in classical (non-relativistic) Mechanics, the natural symmetry group of an isolated mechanical system must contain the symmetry group of the Galilean spacetime, called the Galilean group. This group is of dimension 10. It contains not only the group of Euclidean displacements of space which is of dimension 6 and the group of time translations which is of dimension 1, but the group of linear changes of Galilean reference frames which is of dimension 3.
If we use the Lagrangian formalism or the Hamiltonian formalism, the Lagrangian or the Hamiltonian of the system depends on the reference frame: it is not invariant with respect to linear changes of Galilean reference frames.
It may seem strange to consider the set of all possible motions of a system, which is unknown as long as we have not determined all these possible motions. One may ask if it is really useful when we want to determine not all possible motions, but only one motion with prescribed initial data, since that motion is just one point of the (unknown) manifold of motion! Souriau's answers to this objection are the following. 1.
We know that the manifold of motions has a symplectic structure, and very often many things are known about its symmetry properties. 2. In classical (non-relativistic) mechanics, there exists a natural mathematical object which does not depend on the choice of a particular reference frame (even if the decriptions given to that object by different observers depend on the reference frame used by these observers): it is the evolution space of the system.
The knowledge of the equations which govern the system's evolution allows the full mathematical description of the evolution space, even when these equations are not yet solved.
Moreover, the symmetry properties of the evolution space are the same as those of the manifold of motions.
For example, the evolution space of a classical mechanical system with configuration manifold N is 1. in the Lagrangian formalism, the space R × T N endowed with the presymplectic form d ϖ L , whose kernel is of dimension 1 when the Lagrangian L is hyper-regular, 2. in the Hamiltonian formalism, the space R×T * N with the presymplectic form d ϖ H , whose kernel too is of dimension 1.
The Poincaré-Cartan 1-form ϖ L in the Lagrangian formalism, or ϖ H in the Hamiltonian formalism, depends on the choice of a particular reference frame, made for using the Lagrangian or the Hamiltonian formalism. But their exterior differentials, the presymplectic forms d ϖ L or d ϖ H , do not depend on that choice, modulo a simple change of variables in the evolution space.
Souriau defined this presymplectic form in a framework more general than those of Lagrangian or Hamiltonian formalisms, and called it the Lagrange form. In this more general setting, it may not be an exact 2-form. Souriau proposed as a new Principle, the assumption that it always projects on the space of motions of the systems as a symplectic form, even in Relativistic Mechanics in which the definition of an evolution space is not clear. He called this new principle the Maxwell Principle.
V. Bargmann proved that the symplectic cohomology of the Galilean group is of dimension 1, and Souriau proved that the cohomology class of its action on the manifold of motions of an isolated classical (non-relativistic) mechanical system can be identified with the total mass of the system [55], chapter III, page 153.
Readers interested in the Galilean group and momentum maps of its actions are referred to the recent book by G. de Saxcé and C. Vallée [52].
6 Statistical Mechanics and Thermodynamics

Basic concepts in Statistical Mechanics
During the XVIII-th and XIX-th centuries, the idea that material bodies (fluids as well as solids) are assemblies of a very large number of small, moving particles, began to be considered by some scientists, notably Daniel Bernoulli (1700-1782), Rudolf Clausius (1822-1888), James Clerk Maxwell (1831-1879) and Ludwig Eduardo Boltzmann (1844-1906), as a reasonable possibility. Attemps were made to explain the nature of some measurable macroscopic quantities (for example the temperature of a material body, the pressure exerted by a gas on the walls of the vessel in which it is contained), and the laws which govern the variations of these macroscopic quantities, by application of the laws of Classical Mechanics to the motions of these very small particles. Described in the framework of the Hamiltonian formalism, the material body is considered as a Hamiltonian system whose phase space is a very high dimensional symplectic manifold (M, ω), since an element of that space gives a perfect information about the positions and the velocities of all the particles of the system. The experimental determination of the exact state of the system being impossible, one only can use the probability of presence, at each instant, of the state of the system in various parts of the phase space. Scientists introduced the concept of a statistical state, defined below.  For simplicity we only consider in what follows continuous, very often even smooth statistical states.

Variation in time of a statistical state
Let H be a smooth time independent Hamiltonian on a symplectic manifold (M, ω), X H the associated Hamiltonian vector field and Φ X H its reduced flow. We consider the mechanical system whose time evolution is described by the flow of X H .
If the state of the system at time t 0 , assumed to be perfectly known, is a point z 0 ∈ M, its state at time t 1 is the point z 1 = Φ X H t 1 −t 0 (z 0 ). Let us now assume that the state of the system at time t 0 is not perfectly known, but that a continuous probability measure on the phase space M, whose density with respect to the Liouville measure λ ω is ρ 0 , describes the probability distribution of presence of the state of the system at time t 0 . In other words, ρ 0 is the density of the statistical state of the system at time t 0 . For any other time t 1 , the map Φ X H t 1 −t 0 is a symplectomorphism, therefore leaves invariant the Liouville measure λ ω . The probability density ρ 1 of the statistical state of the system at time t 1 therefore satisfies, for any z 0 ∈ M for which 6.1.6 Definition. Let ρ be the density of a continuous statistical state µ on the symplectic manifold (M, ω). The number is called the entropy of the statistical state µ or, with a slight abuse of language, the entropy of the density ρ.
1. By convention we state that 0 log 0 = 0. With that convention the function x → x log x is continuous on R + . If the integral on the right hand side of the equality which defines s(ρ) does not converge, we state that s(ρ) = −∞. With these conventions, s(ρ) exists for any continuous probability density ρ.

2.
The above definition (6.1.6) of the entropy of a statistical state, founded on ideas developed by Boltzmann in his Kinetic Theory of Gases [9], specially in the derivation of his famous (and controversed) TheoremÊta, is too related with the ideas of Claude Shannon [53] on Information theorey. The use of Information theory in Thermodynamics was more recently proposed by Jaynes [21,22] and Mackey [34]. For a very nice discussion of the use of probability concepts in Physics and application of Information theory in Quantum Mechanics, the reader is referred to the paper by R. Balian [3].
The entropy s(ρ) of a probability density ρ has very remarkable variational properties discussed in the following definitions and proposition.
6.1.8 Definitions. Let ρ be the density of a smooth statistical state on a symplectic manifold (M, ω). 1.
For each function f defined on M, taking its values in R or in some finitedimensional vector space, such that the integral on the right hand side of the equality converges, the value E ρ ( f ) of that integral is called the mean value of f with respect to ρ. 2.
Let f be a smooth function on M, taking its values in R or in some finitedimensional vector space, satisfying the properties stated above. A smooth infinitesimal variation of ρ with fixed mean value of f is a smooth map, defined on the product ] − ε, ε[ ×M, with values in R + , where ε > 0, such that • for τ = 0 and any z ∈ M, ρ(0, z) = ρ(z),

3.
The entropy function s is said to be stationary at the probability density ρ with respect to smooth infinitesimal variations of ρ with fixed mean value of f , if for any smooth infinitesimal variation (τ, z) → ρ(τ, z) of ρ with fixed mean value of f ds(ρ τ ) dτ τ=0 = 0 .
Proof. Let τ → ρ τ be a smooth infinitesimal variation of ρ with fixed mean value of H.
A well known result in calculus of variations shows that the entropy function s is stationary at ρ with respect to smooth infinitesimal variations of ρ with fixed mean value of H, if and only if there exist two real constants a and b, called Lagrange multipliers, such that, for all z ∈ M, 1 + log(ρ) + a + bH = 0 , which leads to ρ = exp(−1 − a − bH) .
By writing that M ρdλ ω = 1, we see that a is determined by b: The following proposition shows that the entropy function, not only is stationary at any Gibbs statistical state, but in a certain sense attains at that state a strict maximum.

be the probability density of the Gibbs statistical state associated to b. We assume that the Hamiltonian H is bounded by below, i.e. that there exists a constant m such that m ≤ H(z) for any z ∈ M. Then the integral defining
converges. For any other smooth probability density ρ 1 such that Proof. Since m ≤ H, the function ρ b exp(−bH) satisfies 0 ≤ ρ b exp(−bH) ≤ exp(−mb)ρ b , therefore is integrable on M. Let ρ 1 be any smooth probability density on M satisfying being convex, its graph is below the tangent at any of its points x 0 , h(x 0 ) . We therefore have, for all x > 0 and x 0 > 0, With x = ρ 1 (z) and x 0 = ρ b (z), z being any element in M, that inequality becomes By integration over M, using the fact that ρ b is the probability density of the Gibbs state associated to b, we obtain We have proven the inequality s( are continuous on M except, maybe, for ϕ, at points z at which ρ b (z) = 0 and ρ 1 (z) = 0, but the set of such points is of measure 0 since ϕ is integrable. They satisfy the inequality ϕ 1 ≤ ϕ. Both are integrable on M and have the same integral. The function ϕ − ϕ 1 is everywhere ≥ 0, is integrable on M and its integral is 0. That function is therefore everywhere equal to 0 on M. We can write, for any z ∈ M, For each z ∈ M such that ρ 1 (z) = 0, we can divide that equality by ρ 1 (z). We obtain Since the function x → x − log x reaches its minimum, equal to 1, for a unique value of x > 0, that value being 1, we see that for each z ∈ M at which ρ 1 (z) > 0, we have ρ 1 (z) = ρ b (z). At points z ∈ M at which ρ 1 (z) = 0, the above equality ( * ) shows that ρ b (z) = 0. Therefore ρ 1 = ρ b .
The following proposition shows that a Gibbs statistical state remains invariant under the flow of the Hamiltonian vector field X H . In that sense, one can say that a Gibbs statistical state is a statistical equilibrium state.

Proposition. Let H be a smooth Hamiltonian bounded by below on a symplectic manifold (M, ω), b ∈ R be such that the integral defining the value P(b) of the partition function P at b converges. The Gibbs state associated to b remains invariant under the flow of of the Hamiltonian vector field X H .
Proof. The density ρ b of the Gibbs state associated to b, with respect to the Liouville measure λ ω , is Since H is constant along each integral curve of X H , ρ b too is constant along each integral curve of X H . Moreover, the Liouville measure λ ω remains invariant under the flow of X H . Therefore the Gibbs probability measure associated to b too remains invariant under that flow.
be the mean value of H with respect to the probability density ρ b . The first and second derivatives with respect to b of the partition function P exist, are continuous functions of b given by

Its derivative with respect to b exists and is a continuous function of b given by
Proof. Using the assumptions 6.

Physical meaning of the introduced functions
Let us consider a physical system, for example a gas contained in a vessel bounded by rigid, thermally insulated walls, at rest in a Galilean reference frame. We assume that its evolution can be mathematically described by means of a Hamiltonian system on a symplectic manifold (M, ω) whose Hamiltonian H satisfies the assumptions 6.2.1. For physicists, a Gibbs statistical state, i.e. a probability measure of density on M, is a thermodynamic equilibrium of the physical system. The set of possible thermodynamic equilibria of the system is therefore indexed by a real parameter b > 0. The following argument will show what physical meaning can have that parameter. Let us consider two similar physical systems, mathematically described by two Hamiltonian systems, of Hamiltonians H 1 on the symplectic manifold (M 1 , ω 1 ) and H 2 on the symplectic manifold (M 2 , ω 2 ). We first assume that they are independent and both in thermodynamic equilibrium, with different values b 1 and b 2 of the parameter b. We denote by E 1 (b 1 ) and E 2 (b 2 ) the mean values of H 1 on the manifold M 1 with respect to the Gibbs state of density ρ 1,b 1 and of H 2 on the manifold M 2 with respect to the Gibbs state of density ρ 2,b 2 . We assume now that the two systems are coupled in a way allowing an exchange of energy. For example, the two vessels containing the two gases can be separated by a wall allowing a heat transfer between them. Coupled together, they make a new physical system, mathematically described by a Hamiltonian system on the symplectic manifold (M1 × M 2 , p * 1 ω 1 + p * 2 ω 2 ), where p 1 : M 1 × M 2 → M 1 and p 2 : M 1 × M 2 → M 2 are the canonical projections. The Hamiltonian of this new system can be made as close to H 1 • p 1 + H 2 • p 2 as one wishes, by making very small the coupling between the two systems. The mean value of the Hamiltonian of the new system is therefore very close to E 1 (b 1 ) + E 2 (b 2 ). When the total system will reach a state of thermodynamic equilibrium, the probability densities of the Gibbs states of its two parts, ρ 1,b ′ on M 1 and ρ 2,b ′ on M 2 will be indexed by the same real number b ′ > 0, which must be such that By 6.2.2, we have, for all b > 0, Therefore b ′ must lie between b 1 and b 2 . If, for example, b 1 < b 2 , we see that . In order to reach a state of thermodynamic equilibrium, energy must be transferred from the part of the system where b has the smallest value, towards the part of the system where b has the highest value, until, at thermodynamic equilibrium, b has the same value everywhere. Everyday experience shows that thermal energy flows from parts of a system where the temperature is higher, towards parts where it is lower. For this reason physicists consider the real variable b as a way to appreciate the temperature of a physical system in a state of thermodynamic equilibrium. More precisely, they state that b = 1 kT where T is the absolute temperature and k a constant depending on the choice of units of energy and temperature, called Boltzmann's constant in honour of the great Austrian scientist Ludwig Eduard Boltzmann (1844Boltzmann ( -1906. For a physical system mathematically described by a Hamiltonian system on a symplectic manifold (M, ω), with H as Hamiltonian, in a state of thermodynamic equilibrium, E(b) and S(b) are the internal energy and the entropy of the system.

Towards thermodynamic equilibrium
Everyday experience shows that a physical system, when submitted to external conditions which remain unchanged for a sufficiently long time, very often reaches a state of thermodynamic equilibrium. At first look, it seems that Lagrangian or Hamiltonian systems with time-independent Lagrangians or Hamiltonians cannot exhibit a similar behaviour. Let us indeed consider a mechanical system whose configuration space is a smooth manifold N, described in the Lagrangian formalism by a smooth time-independent hyper-regular Lagarangian L : T N → R or, in the Hamiltonian formalism, by the associated Hamiltonian Another more serious argument against a kind of thermodynamic behaviour of Lagarangian or Hamiltonian systems rests on the famous recurrence theorem due to H. Poincaré [46]. This theorem asserts indeed that when the useful part of the phase space of the system is of a finite total measure, almost all points in an arbitrarily small open subset of the phase space are recurrent, i.e., the motion starting of such a point at time t 0 repeatedly cross that open subset again and again, infinitely many times when t → +∞.
Let us now consider, instead of perfectly defined states, i.e., points in phase space, statistical states, and ask the question: When at time t = t 0 a Hamiltonian system on a symplectic manifold (M, ω) is in a statistical state given by some probability measure of density ρ 0 with respect to the Liouville measure λ ω , does its statistical state converge, when t → +∞, towards the probability measure of a Gibbs state? This question should be made more precise by specifying what physical meaning has a statistical state and in what mathematical sense a statistical state can converge towards the probability measure of a Gibbs state. A positive partial answer was given by Ludwig Boltzmann when, developing his kinetic theory of gases, he proved his famous (but controversed)Êta theorem stating that the entropy of the statistical state of a gas of small particles is a monotonously increasing function of time. This question, linked with time irreversibility in Physics, is still the subject of important researches, both by physicists and by mathematicians. The reader is referred to the paper [3] by R. Balian for a more thorough discussion of that question.

Examples of thermodynamic equilibria 6.3.1 Classical monoatomic ideal gas
In Classical Mechanics, a dilute gas contained in a vessel at rest in a Galilean reference frame is mathematically described by a Hamiltonian system made by a large number of very small massive particles, which interact by very brief collisions between themselves or with the walls of the vessel, whose motions between two collisions are free. Let us first assume that these particles are material points and that no external field is acting on them, other than that describing the interactions by collisions with the walls of the vessel. The Hamiltonian of one particle in a part of the phase space in which its motion is free is simply where m is the mass of the particle, − → v its velocity vector and − → p its linear momentum vector (in the considered Galilean reference frame), p 1 , p 2 and p 3 the components of − → p in a fixed orhtonormal basis of the physical space.
Let N be the total number of particles, which may not have all the same mass. We use a integer i ∈ {1, 2, . . . , N} to label the particles and denote by m i , − → x i , − → v i , − → p i the mass and the vectors position, velocity and linear momentum of the i-th particle.
The Hamiltonian of the gas is therefore terms involving the collisions between particles and with the walls .
Interactions of the particles with the walls of the vessel are essential for allowing the motions of particles to remain confined. Interactions between particles are essential to allow the exchanges between them of energy and momentum, which play an important part in the evolution with time of the statistical state of the system. However it appears that while these terms are very important to determine the system's evolution with time, they can be neglected, when the gas is dilute enough, if we only want to determine the final statistical state of the system, once a thermodynamic equilibrium is established. The Hamiltonian used will therefore be The partition function is where D is the domain of the 6N-dimensional space spanned by the position vectors − → x i and linear momentum vectors − → p i of the particles in which all the − → x i lie within the vessel containing the gas. An easy calculation leads to where V is the volume of the vessel which contains the gas. The probability density of the Gibbs state associated to b, with respect to the Liouville measure, therefore is We observe that ρ b is the product of the probability densities ρ i,b for the i-th particle The 2N stochastic vectors − → x i and − → p i , i = 1, . . . , N are therefore independent. The position − → x i of the i-th particle is uniformly distributed in the volume of the vessel, while the probability measure of the its linear momentum − → p i is the classical Maxwell-Boltzmann probability distribution of linear momentum for an ideal gas of particles of mass m i , first obtained by Maxwell in 1860. Moreover we see that the three components p i 1 , p i 2 and p i 3 of the linear momentum − → p i in an orhonormal basis of the physical space are independent stochastic variables.
By using the formulae given in 6.2.2 the internal energy E(b) and the entropy S(b) of the gas can be easily deduced from the partition function P(b). Their expressions are We see that each of the N particles present in the gas has the same contribution 3 2b to the internal energy E(b), which does not depend on the mass of the particle. Even more: each degree of freedom of each particle, i.e. each of the the three components of the the linear momentum of the particle on the three axes of an orthonormal basis, has the same contribution 1 2b to the internal energy E(b). This result is known in Physics under the name Theorem of equipartition of the energy at a thermodynamic equilibrium. It can be easily generalized for polyatomic gases, in which a particle may carry, in addition to the kinetic energy due to the velocity of its centre of mass, a kinetic energy due to the particle's rotation around its centre of mass. The reader can consult the books by Souriau [55] and Mackey [34] where the kinetic theory of polyatomic gases is discussed.
The pressure in the gas, denoted by Π(b) because the notation P(b) is already used for the partition function, is due to the change of linear momentum of the particles which occurs at a collision of the particle with the walls of the vessel containing rhe gas (or with a probe used to measure that pressure). A classical argument in the kinetic theory of gases (see for example [13] or [14]) leads to This formula is the well known equation of state of an ideal monoatomic gas relating the number of particles by unit of volume, the pressure and the temperature.
With b = 1 kT , the above expressions are exactly those used in classical Thermodynamics for an ideal monoatomic gas.

Classical ideal monoatomic gas in a gravity field
Let us now assume that the gas, contained in a cylindrical vessel of section Σ and length h, with a vertical axis, is submitted to the vertical gravity field of intensity g directed downwards. We choose Cartesian coordinates x, y, z, the z axis being vertical directed upwards, the bottom of the vessel being in the horizontal surface z = 0. The Hamiltonian of a free particle of mass m, position and linear momentum vectors − → x (components x, y, z) and − → p (components p x , p y and p z ) is As in the previous section we neglect the parts of the Hamiltonian of the gas corresponding to collisions between the particles, or between a particle and the walls of the vessel. The Hamiltonian of the gas is therefore Calculations similar to those of the previous section lead to The expression of ρ b shows that the 2N stochastic vectors − → x i and − → p i still are independent, and that for each i ∈ {1, . . ., N}, the probability law of each stochastic vector − → p i is the same as in the absence of gravity, for the same value of b. Each stochastic vector − → x i is no more uniformly distributed in the vessel containing the gas: its probability density is higher at lower altitudes z, and this nonuniformity is more important for the heavier particles than for the lighter ones. As in the previous section, the formulae given in 6.2.2 allow the calculation of E(b) and S(b). We observe that E(b) now includes the potential energy of the gas in the gravity field, therefore should no more be called the internal energy of the gas.

Relativistic monoatomic ideal gas
In a Galilean reference frame, we consider a relativistic point particle of rest mass m, moving at a velocity − → v . We denote by v the modulus of − → v and by c the modulus of the velocity of light. The motion of the particle can be mathematically described by means of the Euler-Lagrange equations, with the Lagrangian The components of the linear momentum − → p of the particle, in an orthonormal frame at rest in the considered Galilean reference frame, are Denoting by p the modulus of − → p , the Hamiltonian of the particle is Let us consider a relativistic gas, made of N point particles indexed by i ∈ {1, . . . , N}, m i being the rest mass of the i-th particle. With the same assumptions as those made in Section 6.3.1, we can take for Hamiltonian of the gas With the same notations as those of Section 6.3.1, the partition function P of the gas takes the value, for each b > 0, This integral can be expressed in terms of the Bessel function K 2 , whose expression is, for each x > 0, We have This probability density of the Gibbs state shows that the 2N stochastic vectors − → x i and − → p i are independent, that each − → x i is uniformly distributed in the vessel containing the gas and that the probability density of each − → p i is exactly the probability distribution of the linear momentum of particles in a relativistic gas called the Maxwell-Jüttner distribution, obtained by Ferencz Jüttner  in 1911, discussed in the book by the Irish mathematician and physicist J. L. Synge [60]. Of course, the formulae given in 6.2.2 allow the calculation of the internal energy E(b), the entropy S(b) and the pressure Π(b) of the relativistic gas.

Relativistic ideal gas of massless particles
We have seen in the previous chapter that in an inertial reference frame, the Hamiltonian of a relativistic point particle of rest mass m is c p 2 + m 2 c 2 , where p is the modulus of the linear momentum vector − → p of the particle in the considered reference frame. This expression still has a meaning when the rest mass m of the particle is 0. In an orthonormal reference frame, the equations of motion of a particle whose motion is mathematically described by a Hamiltonian system with Hamiltonian which shows that the particle moves on a straight line at the velocity of light c. It seems therefore reasonable to describe a gas of N photons in a vessel of volume V at rest in an inertial reference frame by a Hamiltonian system, with the Hamiltonian With the same notations as those used in the previous section, the partition function P of the gas takes the value, for each b > 0, The probability density of the corresponding Gibbs state, with respect to the Liouville This formula appears in the books by Synge [60] and Souriau [55]. Physicists consider it as not adequate for the description of a gas of photons contained in a vessel at thermal equilibrium because the number of photons in the vessel, at any given temperature, cannot be imposed: it results from the processes of absorption and emission of photons by the walls of the vessel, heated at the imposed temperature, which spontaneously occur. In other words, this number is a stochastic function whose probability law is imposed by Nature. Souriau proposes, in his book [55], a way to account for the possible variation of the number of photons. Instead of using the phase space of the system of N massless relativistic particles contained in a vessel, he uses the manifold of motions M N of that system (which is symplectomorphic to its phase space). He considers that the manifold of motions M of a system of photons in the vessel is the disjoint union of all the manifolds of motions M N of a system of N massless relativistic particles in the vessel, for all possible values of N ∈ N. Fo N = 0 the manifold M 0 is reduced to a singleton with, as Liouville measure, the measure which takes the value 1 on the only non empty part of that manifold (the whole manifold M 0 ). Moreover, since any photon cannot be distinguished from any other photon, two motions of the system with the same number N of massless particles which only differ by the labelling of these particles must be considered as identical. Souriau considers too that since the number N of photons freely adjusts itself, the value of the parameter b = 1 kT must, at thermodynamic equilibrium, be the same in all parts M N of the system, N ∈ N. He uses too the fact that a photon can have two different states of (circular) polarization. With these assumptions the value at any b of the partition function of the system is The number N of photons in the vessel at thermodynamic equilibrium is a stochastic function which takes the value n with the probability The expression of the partition function P allows the calculation of the internal energy, the entropy and all other thermodynamic functions of the system. However, the formula so obtained for the distribution of photons of various energies at a given temperature does not agree with the law, in very good agreement with experiments, obtained by Max Planck (1858-1947) in 1900. An assembly of photons in thermodynamic equilibrium evidently cannot be described as a classical Hamiltonian system. This fact played an important part for the development of Quantum Mechanics.

Specific heat of solids
The motion of a one-dimensional harmonic oscillator can be described by a Hamiltonian system with, as Hamiltonian, The idea that the heat energy of a solid comes from the small vibrations, at a microscopic scale, of its constitutive atoms, lead physicists to attempt to mathematically describe a solid as an assembly of a large number N of three-dimensional harmonic oscillators. By dealing separately with each proper oscillation mode, the solid can even be described as an assembly of 3N one-dimensional harmonic oscillators. Exanges of energy between these oscillators is allowed by the existence of small couplings between them. However, for the determination of the thermodynamic equilibria of the solid we will, as in the previous section for ideal gases, consider as negligible the energy of interactions between the oscillators. We therefore take for Hamiltonian of the solid The value of the paritition function P, for any b > 0, is is the frequency of the i-th harmonic oscillator.
The internal energy of the solid is We observe that it only depends on the the temperature and on the number of atoms in the solid, not on the frequencies ν i of the harmonic oscillators. With b = 1 kT this result is in agreement with the empirical law for the specific heat of solids, in good agreement with experiments at high temperature, discovered in 1819 by the French scientists Pierre Louis Dulong (1785-1838) and Alexis Thérèse Petit (1791-1820).

Generalization for Hamiltonian actions 7.1 Generalized Gibbs states
In his book [56] and in several papers [54,57,58], J.-M. Souriau extends the concept of a Gibbs state for a Hamiltonian action of a Lie group G on a symplectic manifold (M, ω). Usual Gibbs states defined in section 6 for a smooth Hamiltonian H on a symplectic manifold (M, ω) appear as special cases, in which the Lie group is a one-parameter group. If the symplectic manifold (M, ω) is the phase space of the Hamiltonian system, that oneparameter group, whose parameter is the time t, is the group of evolution, as a function of time, of the state of the system, starting from its state at some arbitrarily chosen initial time t 0 . If (M, ω) is the symplectic manifold of all the motions of the system, that oneparameter group, whose parameter is a real τ ∈ R, is the transformation group which maps one motion of the system with some initial state at time t 0 onto the motion of the system with the same initial state at another time (t 0 + τ). We discuss below this generalization.

Notations and conventions
In this section, Φ : G × M → M is a Hamiltonian action (for example on the left) of a Lie group G on a symplectic manifold (M, ω). We denote by G the Lie algebra of G, by G * its dual space and by J : M → G * a momentum map of the action Φ. 7.1.2 Definitions. Let b ∈ G be such that the integrals on the right hand sides of the equalities converge. The smooth probability measure on M with density (with respect to the Liouville measure λ ω on M) is called the generalized Gibbs statistical state associated to b. The functions b → P(b) and b → E J (b) so defined on the subset of G made by elements b for which the integrals defining P(b) and E(J, b) converge are called the partition function associated to the momentum map J and the mean value of J at generalized Gibbs states.
The following Proposition generalizes 6.1.11. 7.1.3 Proposition. Let b ∈ G be such that the integrals defining P(b) and E J (b) in Proposition 7.1.2 converge, and ρ b be the density of the generalized Gibbs state associated to b. The entropy s(ρ b ), which will be denoted by S(b), exists and is given by Moreover, for any other smooth probability density ρ 1 such that 3. Legendre transforms were used by F. Massieu in Thermodynamics in his very early works [40,41], more systematically presented in [42], in which he introduced his characteristic functions (today called thermodynamic potentials) allowing the determination of all the thermodynamic functions of a physical system by partial derivations of a suitably chosen characteristic function. For a modern presentation of that subject the reader is referred to [4] and [11], chapter 5, pp. 131-152. Proof. The orbits of the action on M of the subgroup exp(τb) τ ∈ R of G are the integral curves of the Hamiltonian vector field whose Hamiltonian is J, b , which of course is constant on each of these curves. Therefore the proof of 6.1.12 is valid for that subgroup. are integrable on M with respect to the Liouville measure λ ω ; • moreover their integrals are differentiable with repect to b, their differentials are continuous and can be calculated by differentiation under the sign M .
It is assumed in this section that the considered Hamiltonian action Φ of the Lie group G on the symplectic manifold (M, ω) and its momentum map J are such that the open subset Ω of G is not empty. This condition is not always satisfied when (M, ω) is a cotangent bundle, but of course it is satisfied when it is a compact manifold.
For each b ∈ Ω, the differential at b of the map E J (which is a linear map defined on G, with values in its dual G * ) is given by where we have written, as in 6.1.8, At each b ∈ Ω, the differential of the entropy function S (7.1.3), which is a linear map defined on G, with values in R, in other words an element of G * , is given by Proof. By assumptions 7.2.1, the differentials of P and E J can be calculated by differentiation under the sign M . Easy (but tedious) calculations lead to the indicated results.
Proof. This result follows from the well known result in Probability theory already used in the proof of 6.2.2.
The momentum map J of the Hamiltonian action Φ is not uniquely determined: for any constant µ ∈ G * , J 1 = J + µ too is a momentum map for Φ. The following proposition indicates how the generalized thermodynamic functions P, E J and S change when J is replaced by J 1 .

Proposition.
With the same assumptions and notations as those in Proposition 7.2.2, let µ ∈ G * be a constant. When the momentum map J is replaced by J 1 = J + µ, the open subset Ω of G remains unchanged, while the generalized thermodynamic functions P, E J and S, are replaced, respectively, by P 1 , E J 1 and S 1 , given by The Gibbs satistical state and its density ρ b with respect to the Liouville measure λ ω remain unchanged.
Proof. We have The indicated results follow by easy calculations.
The following proposition indicates how the generalized thermodynamic functions P, E J and S vary along orbits of the adjoint action of the Lie group G on its Lie algebra G. In other words, for each b ∈ Ω and each g ∈ G, Ad g b ∈ Ω. Moreover, let θ : G → G * be the symplectic cocycle of G for the coadjoin action of G on G * such that, for any g ∈ G, Then for each b ∈ Ω and each g ∈ G Proof. We have means that the map E J : Ω → G * is equivariant with respect to the adjoint action of G on the open subset Ω of its Lie algebra G and its affine action on the left on G * (g, ξ ) → Ad * g −1 ξ + θ (g) , g ∈ G , ξ ∈ G * .
where Θ = T e θ : G → G * is the 1-cocycle of the Lie algebra G associated to the 1-cocycle θ of the Lie group G.
Proof. Let us set g = exp(τX ) in the first equality in 7.2.5, derive that equality with respect to τ, and evaluate the result at τ = 0. We obtain Since, by the first equality of 7.2.2, DP(b) = −P(b)E J (b), the first stated equality follows.
Let us now set g = exp(τX ) in the second equality in 7.2.5, derive that equality with respect to τ, and evaluate the result at τ = 0. We obtain the second equality stated.

Corollary.
With the assumptions and notations of 7.2.7, let us define, for each b ∈ Ω, a linear map Θ b : G → G * by setting The map Θ b is a symplectic 1-cocycle of the Lie algebra G for the coadjoint representation, which satisfies Moreover if we replace the momentum map J by J 1 = J + µ, with µ ∈ G * constant, the 1-cocycle Θ b remains unchanged.
Proof. For X , Y and Z in G, we have since Θ is a 1-cocycle, ∑ circ(X,Y,Z) meaning a sum over circular permutations of X , Y and Z, using the Jacobi identity in G, we have ∑ circ(X,Y,Z) The linear map Θ b is therefore a 1 cocycle, even a symplectic 1-cocycle since for all X and Y ∈ G, Using the first equality stated in 7.2.7, we have for any X ∈ G If we replace J by J 1 = J + µ, the map X → Θ(X ) is replaced by X → Θ 1 (X ) = Θ(X ) + ad * X µ and E J (b) by E J 1 (b) = E J (b) + µ, therefore Θ b remains unchanged.
The following lemma will allow us to define, for each b ∈ Ω, a remarkable symmetric bilinear form on the vector subspace [b, G] = [b, X ] ; X ∈ G of the Lie algebra G.

Examples of generalized Gibbs states 7.3.1 Action of the group of rotations on a sphere
The symplectic manifold (M, ω) considered here is the two-dimensional sphere of radius R centered at the origin O of a three-dimensional oriented Euclidean vector space − → E , equipped with its area element as symplectic form. The group G of rotations around the origin (isomorphic to SO(3)) acts on the sphere M by a Hamiltonian action. The Lie algebra G of G can be identified with − → E , the fundamental vector field on M associated to an element − → b in G ≡ − → E being the vector field on M whose value at a point m ∈ M is given by the vector product − → b × − → Om. The dual G * of G will be too identified with − → E , the coupling by duality being given by the Euclidean scalar product. The momentum map J : M → G * ≡ − → E is given by Therefore, for any To calculate the partition function P( − → b ) we choose an orthonormal basis ( − → e x , − → e y , − → e z ) of − → E such that − → b = − → b − → e z , with − → b ∈ R + , and we use angular coordinates (ϕ, θ ) on the sphere M. The coordinates of a point m ∈ M are x = R cos θ cos ϕ , y = R cos θ sin ϕ , z = R sin θ .

We have
The probability density (with respect to the natural area measure on the sphere M) of the generalized Gibbs state associated to − → b is We observe that ρ b reaches its maximal value at the point m ∈ M such that − → Om = R − → b − → b and its minimal value at the diametrally opposed point.

The Galilean group, its Lie algebra and its actions
In view of the presentation, made below, of some physically meaningful generalized Gibbs states for Hamiltonian actions of subgroups of the Galilean group, we recall in this section some notions about the space-time of classical (non-relativistic) Mechanics, the Galilean group, its Lie algebra and its Hamiltonian actions. The interested reader will find a much more detailed treatment on these subjects in the book by Souriau [55] or in the recent book by G. de Saxcé and C. Vallée [52]. The paper [51] presents a nice application of Galilean invariance in Thermodynamics.
The space-time of classical Mechanics is a four-dimensional real affine space which, once an inertial reference frame, units of length and time, orthonormal bases of space and time are chosen, can be identified with R 4 ≡ R 3 × R (coordinates x, y, z, t). The first three coordinates x, y and z can be considered as the three components of a vector − → r ∈ R 3 , therefore an element of space-time can be denoted by ( − → r ,t). However, as the action of the Galilean group will show, the splitting of space-time into space and time is not uniquely determined, it depends on the choice of an inertial reference frame. In classical Mechanics, there exists an absolute time, but no absolute space. There exists instead a space (which is an Euclidean affine three-dimensional space) for each value of the time. The spaces for two distinct values of the time should be considered as disjoint.
The space-time being identified with R 3 × R as explained above, the Galilean group G can be identified with the set of matrices of the form  the vector space R 3 being oriented and endowed with its usual Euclidean structure, the matrix A ∈ SO(3) acting on it.
The action of the Galilean group G on space-time, identified as indicated above with The Lie algebra G of the Galilean group G can be identified with the space of matrices of the form  We have denoted by j( − → ω ) the 3 × 3 skew-symmetric matrix The matrix j( − → ω ) is an element in the Lie algebra so (3), and its action on a vector − → r ∈ R 3 is given by the vector product Let us consider a mechanical system made by a point particle of mass m whose position and velocity at time t, in the reference frame allowing the identification of space-time with R 3 × R, are the vectors − → r and − → v ∈ R 3 . The action of an element of the Galilean group on − → r , − → v and t can be written as  Souriau has shown in his book [55] that this action is Hamiltonian, with the map J, defined on the evolution space of the particle, with value in the dual G * of the Lie algebra G of the Galilean group, as momentum map   be an element in G. Its coupling with J( − → r ,t, − → v , m) ∈ G * is given by the formula

One-parameter subgroups of the Galilean group
In his book [55], J.-M. Souriau has shown that when the considered Lie group action is the action of the full Galilean group on the space of motions of an isolated mechanical system, the open subset Ω of the Lie algebra G of the Galilean group on which the conditions specified in 7.2.1 are satisfied is empty. In other words, generalized Gibbs states of the full Galilean group do not exist. However, generalized Gibbs states for one-parameter subgroups of the Galilean group do exist which have an interesting physical meaning. Let us consider the element b of G given by formula ( * ) of 7.3.2, and assume that ε = 0. The one-parameter subgroup G 1 of the Galilean group generated by b is the set of matrices exp(τb), with τ ∈ R. We have with the usual convention that j( − → ω ) 0 is the unit matrix.
The physical meaning of this one-parameter subgroup of the Galilean group can be understood as follows. Let us call fixed the affine Euclidean reference frame of space The above equality ( * ) is well suited for the determination of generalized Gibbs states of the system. Let us set The integrals in the right hand sides of these equalities converge if and only if ε < 0. It means that the matrix b belongs to the subset Ω of the one-dimensional Lie algebra of the considered one-parameter subgroup G 1 of the Galilean group on which generalized Gibbs states can be defined if and only if ε < 0. Assuming that condition satisfied, we can use Definitions 7.1.2. The generalized Gibbs state determined by b has the smooth density, with respect to the Liouville measure ∏ N i=1 λ ω i on the symplectic manifold of motions The partition function, whose expression is The length ∆ = − → e z × − → r i,0 is the distance between the i-th particle and the axis of rotation of the moving frame (the coordinate z axis). Moreover, we have seen that ε = −1 kT . Therefore in the generalized Gibbs state, the probability density ρ i (b) of presence of the i-th particle in its symplectic manifold of motion M i , ω i , with respect to the Liouville measure λ ω i , is This formula describes the behaviour of a gas made of point particles of various masses in a centrifuge rotating at a constant angular velocity ω ε : the heavier particles concentrate farther from the rotation axis than the lighter ones.

Other applications of generalized Gibbs states
Applications of generalized Gibbs states in Thermodynamics of Continua, with the use of affine tensors, are presented in the papers by G. de Saxcé [49,50].
Several applications of generalized Gibbs states of subgroups of the Poincaré group were considered by J.-M. Souriau. For example, he presents in his book [55], chapter IV, page 308, a generalized Gibbs which describes the behaviour of a gas in a relativistic centrifuge, and in his papers [56,57], very nice applications of such generalized Gibbs states in Cosmology.