From Tools in Symplectic and Poisson Geometry to J.-M. Souriau’s Theories of Statistical Mechanics and Thermodynamics ^†

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 Section 2 and Section 3, the Hamiltonian formalism and symmetries of Hamiltonian systems in Section 4 and Section 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 [1,2,3,4,5,6,7,8,9,10,11] discuss, much more fully than in the present paper, the contents of Section 2, Section 3, Section 4 and Section 5. The interested reader is referred to these books for detailed proofs of results whose proofs are only briefly sketched here. The recent paper [12] contains detailed proofs of most results presented here in Section 4 and Section 5.

The main sources used for Section 6 and Section 7 are the book and papers by Jean-Marie Souriau [13,14,15,16,17] and the beautiful small book by Mackey [18].

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.

The notations used are more or less those generally used now in differential geometry. The tangent and cotangent bundles to a smooth manifold M are denoted by $TM$ and $T^{*}M$, respectively, and their canonical projections by ${\tau }_M:TM\to M$ and ${\pi }_M:T^{*}M\to M$. The vector spaces of k-multivectors and k-forms on M are denoted by $A^k(M)$ and ${\mathsf{\Omega }}^k(M)$, respectively, with $k\in \mathbb{Z}$ and, of course, $A^k(M)=\left\{0\right\}$ and ${\mathsf{\Omega }}^k(M)=\left\{0\right\}$ if $k<0$ and if $k>dimM$, k-multivectors and k-forms being skew-symmetric. The exterior algebras of multivectors and forms of all degrees are denoted by $A(M)={\oplus }_k{A}^k(M)$ and $\mathsf{\Omega }(M)={\oplus }_k{\mathsf{\Omega }}^k(M)$, respectively. The exterior differentiation operator of differential forms on a smooth manifold M is denoted by $\mathrm{d}:\mathsf{\Omega }(M)\to \mathsf{\Omega }(M)$. The interior product of a differential form $\eta \in \mathsf{\Omega }(M)$ by a vector field $X\in A^1(M)$ is denoted by $\mathrm{i}(X)\eta$.

Let $f:M\to N$ be a smooth map defined on a smooth manifold M, with values in another smooth manifold N. The pull-back of a form $\eta \in \mathsf{\Omega }(N)$ by a smooth map $f:M\to N$ is denoted by $f^{*}\eta \in \mathsf{\Omega }(M)$.

A smooth, time-dependent vector field on the smooth manifold M is a smooth map $X:\mathbb{R}\times M\to TM$ such that, for each $t\in \mathbb{R}$ and $x\in M$, $X(t,x)\in T_{x}M$, the vector space tangent to M at x. When, for any $x\in M$, $X(t,x)$ does not depend on $t\in \mathbb{R}$, 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 $\mathbb{R}\times M$ instead than on the whole $\mathbb{R}\times M$. It defines a differential equation said to be associated to X. The (full) flow of X is the map ${\mathsf{\Psi }}^X$, defined on an open subset of $\mathbb{R}\times \mathbb{R}\times M$, taking its values in M, such that for each $t_0\in \mathbb{R}$ and $x_0\in M$ the parametrized curve $t\mapsto {\mathsf{\Psi }}^X(t,t_0,x_0)$ is the maximal integral curve of Equation (1) satisfying $\mathsf{\Psi }(t_0,t_0,x_0)=x_0$. When $t_0$ and $t\in \mathbb{R}$ are fixed, the map $x_0\mapsto {\mathsf{\Psi }}^X(t,t_0,x_0)$ 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 ${\mathsf{\Psi }}_{(t,t_0)}^X$. When X is in fact a vector field in the usual sense (not dependent on time), ${\mathsf{\Psi }}_{(t,t_0)}^X$ only depends on $t-t_0$. Instead of the full flow of X we can use its reduced flow ${\mathsf{\Phi }}^X$, defined on an open subset of $\mathbb{R}\times M$ and taking its values in M, related to the full flow ${\mathsf{\Psi }}^X$ by

For each $t\in \mathbb{R}$, the map $x_0\mapsto {\mathsf{\Phi }}^X(t,x_0)={\mathsf{\Psi }}^X(t,0,x_0)$ is a diffeomorphism, denoted by ${\mathsf{\Phi }}_t^X$, defined on an open subset of M (which may be empty) onto another open subset of M.

When $f:M\to N$ is a smooth map defined on a smooth manifold M, with values in another smooth manifold N, there exists a smooth map $Tf:TM\to TN$ called the prolongation of f to vectors, which for each fixed $x\in M$ linearly maps $T_{x}M$ into $T_{f(x)}N$. When f is a diffeomorphism of M onto N, $Tf$ is an isomorphism of $TM$ onto $TN$. That property allows us to define the canonical lifts of a vector field X in $A^1(M)$ to the tangent bundle $TM$ and to the cotangent bundle $T^{*}M$. Indeed, for each $t\in \mathbb{R}$, ${\mathsf{\Phi }}_t^X$ is a diffeomorphism of an open subset of M onto another open subset of M. Therefore $T{\mathsf{\Phi }}_t^X$ is a diffeomorphism of an open subset of $TM$ onto another open subset of $TM$. It turns out that when t takes all possible values in $\mathbb{R}$ the set of all diffeomorphisms $T{\mathsf{\Phi }}_t^X$ is the reduced flow of a vector field $\overline{X}$ on $TM$, which is the canonical lift of X to the tangent bundle $TM$.

Similarly, the transpose ${(T{\mathsf{\Phi }}_{-t}^X)}^T$ of $T{\mathsf{\Phi }}_{-t}^X$ is a diffeomorphism of an open subset of the cotangent bundle $T^{*}M$ onto another open subset of $T^{*}M$, and when t takes all possible values in $\mathbb{R}$ the set of all diffeomorphisms ${(T{\mathsf{\Phi }}_{-t}^X)}^T$ is the reduced flow of a vector field $\widehat{X}$ on $T^{*}M$, which is the canonical lift of X to the cotangent bundle $T^{*}M$.

The canonical lifts of a vector field to the tangent and cotangent bundles are used in Section 3 and Section 5. They can be defined too for time-dependent vector fields.

The principles of mechanics were stated by the great English mathematician Isaac Newton (1642–1727) in his book Philosophia Naturalis Principia Mathematica published in 1687 [19]. On this basis, a little more than a century later, Joseph Louis Lagrange (1736–1813) in his book Mécanique analytique [20] 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 $TN$ 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 $x^1,\dots ,x^n$ of a point in N, and a kinematic state by the $2n$ coordinates $x^1,\dots ,x^n,v^1,\dots v^n$ of a vector tangent to N at some element in N.

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 $\mathbb{R}$ (spanned by the variable t representing the time) with the manifold $TN$ of kinematic states of the system. In local coordinates, the Lagrangian L is expressed as a function of the $2n+1$ variables, $t,x^1,\dots ,x^n,v^1,\dots ,v^n$ and the Euler–Lagrange equations have the remarkably simple form where $x(t)$ stands for $x^1(t),\dots ,x^n(t)$ and $v(t)$ for $v^1(t),\dots ,v^n(t)$ with, of course,

The great Irish mathematician William Rowan Hamilton (1805–1865) observed [21,22] 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 (Lagrange observed that fact before Hamilton, but in the last edition of his book he chose to derive the Euler–Lagrange equations by application of the principle of virtual works, using a very clever evaluation of the virtual work of inertial forces for a smooth infinitesimal variation of the motion). where $\gamma :[t_0,t_1]\to N$ is a smooth curve in N parametrized by the time t. These equations express the fact that the action integral $I_L(\gamma )$ is stationary with respect to any smooth infinitesimal variation of γ with fixed end-points $\left(t_0,\gamma (t_0)\right)$ and $\left(t_1,\gamma (t_1)\right)$. 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 [23,24,25].

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 mathematician Élie Cartan (1869–1951). Let us introduce the concepts used by the statement of this theorem.

The interested reader will find the proof of that theorem in [26], (Theorem 2.2, Chapter IV, p. 262) or, for hyper-regular Lagrangians (an additional assumption which in fact, is not necessary) in [27], Chapter IV, Theorem 2.1, p. 167.

In this section N is the configuration space of a conservative Lagrangian mechanical system with a smooth, maybe time dependent Lagrangian $L:\mathbb{R}\times TN\to \mathbb{R}$. Let ${\widehat{\varpi }}_L$ be the Poincaré-Cartan 1-form on the evolution space $\mathbb{R}\times TN$.

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.

Since Z is an infinitesimal symmetry of ${\widehat{\varpi }}_L$,

Using the well known formula relating the Lie derivative, the interior product and the exterior derivative we can write ☐

The Lie bracket of two infinitesimal symmetries of ${\widehat{\varpi }}_L$ is too an infinitesimal symmetry of ${\widehat{\varpi }}_L$. Let us therefore assume that there exists a finite-dimensional Lie algebra of vector fields on $\mathbb{R}\times TN$ whose elements are infinitesimal symmetries of ${\widehat{\varpi }}_L$.

In a short Note [29] published in 1901, the great french mathematician Henri Poincaré (1854–1912) proposed a new formulation of the equations of mechanics.

Let N be the configuration manifold of a conservative Lagrangian system, with a smooth Lagrangian $L:TN\to \mathbb{R}$ which does not depend explicitly on time. Poincaré assumes that there exists an homomorphism ψ of a finite-dimensional real Lie algebra $\mathcal{G}$ into the Lie algebra $A^1(N)$ of smooth vector fields on N, such that for each $x\in N$, the values at x of the vector fields $\psi (X)$, when X varies in $\mathcal{G}$, completely fill the tangent space $T_{x}N$. The action ψ is then said to be locally transitive.

Of course these assumptions imply $dim\mathcal{G}\ge dimN$.

Under these assumptions, Henri Poincaré proved that the equations of motion of the Lagrangian system could be written on $N\times \mathcal{G}$ or on $N\times {\mathcal{G}}^{*}$, where ${\mathcal{G}}^{*}$ is the dual of the Lie algebra $\mathcal{G}$, instead of on the tangent bundle $TN$. When $dim\mathcal{G}=dimN$ (which can occur only when the tangent bundle $TN$ 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 $dim\mathcal{G}>dimN$, the system made by the Euler–Poincaré equation is underdetermined.

Let $\gamma :[t_0,t_1]\to N$ be a smooth parametrized curve in N. Poincaré proves that there exists a smooth curve $V:[t_0,t_1]\to \mathcal{G}$ in the Lie algebra $\mathcal{G}$ such that, for each $t\in [t_0,t_1]$,

When $dim\mathcal{G}>dimN$ the smooth curve V in $\mathcal{G}$ is not uniquely determined by the smooth curve γ in N. However, instead of writing the second-order Euler–Lagrange differential equations on $TN$ 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 (2), if and only if γ is a possible motion of the Lagrangian system.

Let $\phi :N\times \mathcal{G}\to TN$ and $\overline{L}:N\times \mathcal{G}\to \mathbb{R}$ be the maps

We denote by ${\mathrm{d}}_1\overline{L}:N\times \mathcal{G}\to T^{*}N$ and by $d_2\overline{L}:N\times \mathcal{G}\to {\mathcal{G}}^{*}$ the partial differentials of $\overline{L}:N\times \mathcal{G}\to \mathbb{R}$ with respect to its first variable $x\in N$ and with respect to its second variable $X\in \mathcal{G}$.

The map $\phi :N\times \mathcal{G}\to TN$ is a surjective vector bundles morphism of the trivial vector bundle $N\times \mathcal{G}$ into the tangent bundle $TN$. Its transpose ${\phi }^T:T^{*}N\to N\times {\mathcal{G}}^{*}$ is therefore an injective vector bundles morphism, which can be written where ${\pi }_N:T^{*}N\to N$ is the canonical projection of the cotangent bundle and $J:T^{*}N\to {\mathcal{G}}^{*}$ 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 $X\mapsto \phi (x,X)=\psi (X)(x)$.

Let ${\mathcal{L}}_L={\mathrm{d}}_{\mathrm{vert}}L:TN\to T^{*}N$ be the Legendre map defined in Definition 1.

The interested reader will find a proof of that theorem in local coordinates in the original Note by Poincaré [29]. More intrinsic proofs can be found in [12,30]. Another proof is possible, in which that theorem is deduced from the Euler-Cartan Theorem 1.

Examples of applications of the Euler–Poincaré equation can be found in [5,6,12,30] and, for an application in thermodynamics, [31].

Let M be a manifold endowed with some structure, which can be either

• a presymplectic structure, determined by a presymplectic form, i.e., a 2-form ω which is closed ($\mathrm{d}\omega =0$),
• a symplectic structure, determined by a symplectic form ω, i.e., a 2-form ω which is both closed ($\mathrm{d}\omega =0$) and nondegenerate ($ker\omega =\left\{0\right\}$),
• a Poisson structure, determined by a smooth Poisson bivector field Λ satisfying $[\Lambda ,\Lambda ]=0$.

Not any smooth function on a presymplectic manifold can be a Hamiltonian.

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 [8,9,10].

The proof of that result, which is easy, can be found for example in [8,9,10].