F ◦Φ^{s}_{G}= e^{sL}^{G}F . (1.21)

CPROOF. Set ˆF(s) := F ◦Φ_{G}^{s}, and observe that G ◦Φ_{G}^{s} = G (why?). ThenF = { ˆ˙ˆ F, G} = LGF,ˆ
F = {¨ˆ F, G} = {{ ˆ˙ˆ F, G}, G} = L^{2}_{G}F and so on. Thus, the Taylor expansion of ˆˆ F(s) centered at s = 0
reads

F ◦Φ_{G}^{s} = ˆF(s) = F + sLGF +s^{2}

2L^{2}_{G}F + ··· = e^{sL}^{G}F . B

**1.3** **Integrability: Liouville theorem**

A dynamical system is integrable if it possesses a number of first integrals (i.e. functions
defined on the phase space not evolving in time along the flow of the system) which is high
enough to geometrically constraint the motion, a priori, on a curve. For a generic system of the
form ˙x = u(x) in R^{n}, integrability would require, a priori, n−1 first integrals (the intersection of
the level sets of m first integrals has co-dimension m and dimension n − m). However, it turns
out that the Hamiltonian structure reduces such a number to half the (even) dimension of the
phase space.

**Definition 1.3. The system defined by the Hamiltonian H(q, p, t), is said to be integrable in**
Γ⊆ R^{2n}, in the sense of Liouville, if it admitsn independent first integrals f_{1}(q, p, t), . . . , f_{n}(q, p, t)
in involution, i.e., for any (q, p) ∈Γ^{and}t ∈ R

1. *∂f*j/*∂t + {f*j, H} = 0 for any j = 1,..., n;

2. Pn

j=1c_{j}∇ fj(q, p, t) = 0 ⇒ c1= · · · = cn= 0 (equivalently: the rectangular matrix of the
gra-dients of the integrals has maximal rankn) for any (q, p, t);

3. { fj, f_{k}} = 0 for any j, k = 1,..., n.

Notice that often H coincides with one of the first integrals. The introduction of the above definition is motivated by the following theorem.

**Theorem 1.1 (Liouville). Let the Hamiltonian system defined by H be Liouville-integrable in**
Γ⊆ R^{2n}, and leta ∈ R^{n}such that the level set

M_{a}:= {(q, p) ∈Γ ^{: f}1(q, p, t) = a1, . . . , fn(q, p, t) = an}

is non empty; let also M^{0}_{a} denote a (nonempty) connected component of M_{a}. Then, a
func-tion S(q, t; a) exists, such that p · dq|Ma^{0} = dqS(q, a) and S is a complete integral of the
time-dependent Hamilton-Jacobi equation, i.e. the generating function of a canonical transformation
C ^{: (q}*, p, H, t) 7→ (b, a,0, t), where b := ∂S/∂a.*

**Remark 1.2. If H(q, p) does not depend explicitly on time, then in the above definition of**
integrable system all the f_{j} are independent of time as well, and condition 1. is replaced by
{ f_{j}, H} = 0. In such a case, the generating function S(q; a) appearing in the Liouville theorem is
a complete integral of the time-independent Hamilton-Jacobi equation H(q, ∇aS) = K(a), thus
generating a canonical transformationC ^{: (q}, p) 7→ (b, a) such that H(C^{−1}^{(b}, a)) = K(a).

In order to better understand the meaning of the definition 1.3 and to prove the theorem 1.1,
we start by supposing that H(q, p, t) admits n independent first integrals f_{1}(q, p, t), . . . , f_{n}(q, p, t),
but we do not suppose, for the moment, that such first integrals are in involution. Without any
loss of generality, as a condition of independence of the first integrals one can assume

det repre-sented, by means of the implicit function theorem, as

p_{1}= u1(q, t; a) ; . . . p_{n}= un(q, t; a) . (1.22)
The above relations must hold at any time if they hold at t = 0. Differentiating the relation
p_{i}(t) = ui(q(t), t; a) (i = 1,..., n) with respect to time and using the Hamilton equations one gets

Notice that, for the sake of convenience, the same sum of terms is artificially added on both sides of the equation. By introducing the quantities

rot(u) := the equations (1.23) can be rewritten in compact, vector form as

*∂u*

*∂t* + rot(u)v = −∇qH .e (1.27)

Notice the similarity of the latter equation with the (unitary density) Euler equation of hydro-dynamics, namely

* where u is the velocity field, p is the pressure and rot(u)u = ω∧u, ω := ∇∧u being the vorticity*
of the fluid. The similarity of (1.27) and (1.28) is completely evident in the case of natural
mechanical systems, whose Hamiltonian has the form

H(q, p, t) = p · G(q, t)p

2 + V (q, t) ,

where G(q, t) is a n×n positive definite matrix. In such a case v = Gu and equation (1.27) takes the rather simple form

In particular, in those cases such that G = Inthe latter equation is the Euler equation in space dimension n, with the potential energy V playing the role of pressure.

1.3. INTEGRABILITY: LIOUVILLE THEOREM 17
**Remark 1.3. Attention has to be paid to the fact that for the Euler equation (1.28) the pressure p**
**is determined by the divergence-free condition ∇· u = 0, while nothing similar holds, in general,**
for the equations (1.27) or (1.29).

Now, by analogy with the case of fluids, we look for curl-free, i.e. irrotational solutions of the
Euler-like equation (1.27) (we recall that in fluid dynamics, looking for a solution of the Euler
**equation (1.28) of the form u = ∇φ leads to the Bernoulli equation for the velocity potential***φ, namely ∂φ/∂t + |∇φ|*^{2}/2 + p = constant). In simply connected domains (of the n-dimensional
configuration space), one has

rot(u) = 0 iff u = ∇S ,

where S = S(q, t; a). Upon substitution of u = ∇S into equation (1.27) and lifting a gradient, one gets

*∂S*

*∂t* + H(q, ∇q*S, t) = ϕ(t; a) .* (1.30)

One can set*ϕ(t; a) ≡ 0 without any loss of generality, and the latter equation becomes the *
time-dependent Hamilton-Jacobi equation (if*ϕ 6≡ 0 then*S := S −e R

*ϕdt satisfies equation (1.30) with*
zero right hand side). Thus, The Hamilton-Jacobi equation is the analogue of the Bernoulli
equation for the hydrodynamics of Hamiltonian systems. The interesting point is that the
curl-free condition rot(u) = 0 is equivalent to the condition of involution of the first integrals

f_{1}, . . . , f_{n}. Indeed, starting from the identity

f_{i}(q, u(q, t; a), t) ≡ ai , (1.31)
and taking its derivative with respect to q_{j} one gets

*∂f*i
the reverse one requires the independence condition det(*∂f /∂p) 6= 0). This is the key point: the*
condition of involution of the first integrals is equivalent to that of irrotational, i.e. gradient,
velocity fields of the hydrodynamic equation (1.27). The velocity potential S(q, t; a) satisfies the
Hamilton-Jacobi equation and is actually a complete integral of the latter. In order to see this,
one can start again from identity (1.31), setting there u = ∇S and taking the derivative with
respect to aj, getting the i, j component of the matrix identity

µ*∂f*

which, by the independence condition of the first integrals, yields det(*∂*^{2}S/*∂q∂a) 6= 0. We finally*
notice that if the first integrals and thus the velocity field u are known, then the potential S
can be obtained by a simple integration, based on the identity d_{q}S = u · dq, such as

S(q, t; a) − S(0, t; a) = Z

0→q

u(q^{0}, t; a) · dq^{0}=
Z _{1}

0

u(*λq, t; a) · qdλ ,*

where S(0, t; a) may be set to zero. The function S(q, t; a), satisfying the Hamilton-Jacobi
equa-tion, generates a canonical transformation (q, p, H, t) 7→ (b, a,0, t) to a zero Hamiltonian,
trans-formation defined by the implicit equations p = ∇qS(q, t; a), b := ∇aS(q, t; a). What just
re-ported above is actually the proof of theorem 1.1. The restriction to the case where H, f_{1}, . . . , f_{n}
are independent of time is left as an exercise.

**Example 1.8. The Hamiltonian system of central motions is Liouville-integrable. Indeed, if**
H = ^{|p|}_{2m}^{2} **+ V (|r|) is the Hamiltonian of the system, then it is easily proven that the angular**
**momentum L = r ∧ p is a vector constant of motion (the Hamiltonian is invariant with respect**
**to the “canonical rotations” (r, p) 7→ (r**^{0}**, p**^{0}**) = (Rr, R p), where R is any orthogonal matrix; the**
conservation of the angular momentum is a consequence of the Nöther theorem). The phase
space of the system has dimension 2n = 6, and three independent first integrals in involution
are f_{1}:= H, f2**:= |L|**^{2} and f_{3}:= Lz, for example (show that).

**Example 1.9. The Hamiltonian of n noninteracting systems, H =**P_{n}

j=1h_{j}(q_{j}, p_{j}), is obviously
Liouville integrable, with the choice f_{j}:= hj(q_{j}, p_{j}), j = 1,..., n. As an example, consider the
case of harmonic oscillators, whereh_{j}(q_{j}, p_{j}) = (p^{2}_{j}*+ ω*^{2}_{j}q^{2}_{j})/2.

**Example 1.10. The wave equation u**_{tt} = uxx is an infinite dimensional Liouville integrable
Hamiltonian system. For the sake of simplicity we consider the case of fixed ends: u(0, t) =
*0 = u(`, t), u*t(0, t) = 0 = ut(*`, t). As previously shown, the equation has the obvious *
Hamilto-nian form u_{t}*= p = δH/δp, p*t= uxx*= −δH/δu, where H(u, p) =*R_{`}

0

p^{2}+(ux)^{2}

2 dx. Since the set of
functions*ϕ*k(x) :=p

2/*`sin(πkx/`), k ≥ 1, is an orthonormal basis in the Hilbert space L*2([0,*`])*
of square integrable functions on [0,*`] with fixed ends, one can expand both u(x, t) and p(x, t)*
in Fourier series: u(x, t) =P

k≥1uˆ_{k}(t)*ϕ*k(x), p(x, t) =P

k≥1pˆ_{k}(t)*ϕ*k(x), with Fourier coefficients
given by ˆu_{k}(t) =R_{`}

0 u(x, t)*ϕ*k(x)dx and ˆp_{k}(t) =R_{`}

0 p(x, t)*ϕ*k(x)dx, respectively. Upon substitution
of the latter Fourier expansion into the Hamiltonian wave equation one easily gets ˙ˆu_{k}= ˆpk,

˙ˆp_{k}*= −ω*^{2}_{k}uˆ_{k}, where *ω*k *:= (πk/`)*^{2}. These are the Hamilton equations of a system of infinitely
many harmonic oscillators, with HamiltonianK =P

k≥1( ˆp^{2}_{k}*+ ω*^{2}_{k}qˆ^{2}_{k})/2, which is obviously
Liou-ville integrable with the choicef_{k}= ( ˆp^{2}_{k}*+ω*^{2}_{k}qˆ^{2}_{k})/2, k ≥ 1. One easily finds that the substitution of
the Fourier expansions ofu and p into the wave equation Hamiltonian H yields H = K (to such
a purpose, notice thatR_{`}

0(u_{x})^{2}dx = uux|^{`}_{0}−R_{`}

0 uu_{xx}dx).

Useful reference books for the present chapter are [6] and [27]. Hydrodynamics of Hamil-tonian system is originally discussed e.g. in paper [24].

**Chapter 2**

**Probabilistic approach**

In many cases, instead of trying to control the details of the dynamics of a given system, it is convenient to approach the problem from the point of view of probability theory, trying to characterize the statistical aspects of the dynamics itself. To such a purpose, the phase space of the system has to be endowed with a probability measure that does not evolve along the flow, so that mean values of observables are independent of time. One of the most important results of such an approach is the deduction of the macroscopic laws of thermodynamics for mechanical systems with many degrees of freedom.

**2.1** **Probability measures and integration**

Given a setΩ(think e.g. to a differentiable manifold) let us denote by 2^{Ω} the power set ofΩ^{,}
i.e. the set of all, proper and improper, subsets ofΩ(recall that the the notation is due to the
fact that for a finite set of s elements the dimension of its power set is 2^{s}).

**Definition 2.1. A set***σ*Ω⊆ 2^{Ω} is called a*σ-algebra on*Ω^{if}
1. it containsΩ^{;}

2. it is closed with respect to complementation, i.e. *A ∈ σ*_{Ω}⇒ A^{c}*∈ σ*_{Ω};
3. it is closed with respect to countable union, i.e. {A_{j}}_{j∈N}*∈ σ*_{Ω}⇒S

j∈NA_{j}*∈ σ*_{Ω}.

Notice that the complement of a countable union of sets is the countable intersection of the complements of those sets, which means that closure with respect to complementation and countable union implies closure with respect to countable intersection.

Due to the fact that 2^{Ω}is a*σ-algebra and the intersection of σ-algebras is still a σ-algebra,*
ifF ⊂ 2^{Ω} denotes a set of subsets ofΩ, the smallest*σ-algebra containing*F always exists and
is usually denoted by *σ*_{Ω}(F ), which is also refereed to as the σ-algebra generated by F . In
this respect, ifΩis endowed with a topology, a*σ-algebra particularly relevant to applications*
is the one generated by the open sets ofΩ, which is called the Borel*σ-algebra of*Ω^{.}

**Definition 2.2. Given a set**Ωand a sigma-algebra*σ*_{Ω} on ti, a probability measure on Ω^{is a}
nonnegative function*µ : σ*_{Ω}→ [0, 1] which is

19

• normalized, i.e.*µ(*Ω) = 1;

• countably additive, i.e. additive with respect to countable unions of pairwise disjoint sets:

{A_{j}}_{j∈N}*∈ σ*_{Ω} andA_{i}∩ Aj*= ; ∀i 6= j ⇒ µ(*S

j∈NA_{j}) =P

j∈N*µ(A*j).

A set A ⊂Ωis then said to be *µ-measurable if A ∈ σ*_{Ω}. Moreover, if A is measurable and
*µ(A) = 0, then any set B ⊂ A is assumed to have measure zero. The general additivity law is*
readily proven by observing that A \ (A ∩ B), B \ (A ∩ B) and A ∩ B are pairwise disjoint sets
whose union yields A ∪ B. Moreover, A \ B and A ∩ B are disjoint sets, their union being A, so
that*µ(A \ B) = µ(A) − µ(A ∩ B). Thus, one gets*

*µ(A ∪ B) = µ(A) + µ(B) − µ(A ∩ B) ≤ µ(A) + µ(B) ,* (2.1)
the equality sign holding iff A ∩ B has measure zero, which is true, in particular, when the
intersection is empty.

**Definition 2.3. A property relative to the elements** *ω ∈*Ωis said to hold *µ-almost everywhere*
(in short*µ-a.e.) in the measurable set A ⊆*Ω*if it holds ∀ω ∈ A \ B and B has measure zero.*

If Ω is finite or countably infinite, one can always build up a probability measure on the
largest *σ-algebra 2*^{Ω}, in the natural way, by assigning a function p :Ω*→ [0, 1] : ω 7→ p** _{ω}* such
that P

*ω∈Ω*p* _{ω}*= 1. Indeed, Given A ∈ 2

^{Ω}, since A =S

*ω∈A*{*ω}, then, by the countable additivity*
of the measure*µ one has*

*µ(A) = µ*
Ã

[

*ω∈A*

{*ω}*

!

= X

*ω∈A**µ({ω}) .*

Thus, the measure *µ(A) of any measurable set A is completely determined by the value of*
the measure of all its singletons (i.e. subsets consisting of a single element), and one has to
assign p_{ω}*:= µ({ω}). The normalization of the sum of the p** _{ω}*’s follows taking A =Ωin the above
displayed equation and using

*µ(*Ω) = 1. IfΩis uncountable, the latter procedure does not work, in general.

**Example 2.1. Consider the case**Ω*:= [0,1]. A natural (probability) measure µ on*Ω^{should be}
*such that if 0 ≤ a ≤ b ≤ 1, then µ([a, b]) = b − a. Observe that the singletons are the set consisting*
of a single point*ω of*Ω, and that by shrinking any interval to a single point one gets *µ({ω}) = 0*

*∀ω ∈*Ω. Thus, one cannot define such a natural measure on singletons. Moreover, if one tries to
define the candidate measure at hand on the uncountable power set 2^{Ω}, it can be proven that no
such measure exists: the power set is too large.

With a (probability) measure*µ on*Ω, one defines an integration overΩas follows. First of
all, if*χ*A denotes the characteristic (or indicator) function of the measurable set A (*χ*A(x) = 1 if
x ∈ A and zero otherwise), one defines

Z

A

d*µ =*
Z

Ω*χ*Ad*µ := µ(A) ;*
Z

B*χ*A d*µ :=*

Z

A∩B

d*µ =*
Z

Ω*χ*_{A∩B} d*µ = µ(A ∩ B) .*

2.1. PROBABILITY MEASURES AND INTEGRATION 21 In this way one can define the integration of the so-called simple functions, namely functions that are (finite) linear combinations of characteristic functions of given sets. Thus, if S = P

More general functions are then approximated through sequences of simple functions. More precisely, if F ≥ 0, one sets

For a function F with non constant sign one then introduces the positive part F^{+}:= max{0, F}

and negative part F^{−}= max{0, −F} = −min{0, F} of F (notice that both F^{+} and F^{−} are
nonneg-ative by definition).

**Definition 2.4. A function F is said to be integrable over B ⊆**Ωwith respect to the measure*µ*
if

Notice that the latter definition of integrability is equivalent to require that bothR

BF^{±} d*µ*

**Definition 2.5. The space of integrable functions over**Ωwith respect to*µ is denoted by*L1(Ω^{,}*µ).*

In general, for any p ≥ 1 one defines

Lp(Ω^{,}*µ) :=*

**Definition 2.6. Given two probability measures***µ and ν on*Ω(i.e. defined on the same*σ*_{Ω}),*µ*
is said to be absolutely continuous with respect to*ν if for any set A such that ν(A) = 0 it follows*
*µ(A) = 0.*

By the Radon-Nikodym theorem, if*µ is absolutely continuous with respect to ν, then µ has*
a density, namely there exists a nonnegative*ν-integrable function % :*Ω→ R^{+}such that

*µ(A) =*
referring to the latter as the Radon-Nikodym derivative of*µ with respect to ν.*

The most relevant case in applications is that of measures absolutely continuous with re-spect to the Lebesgue measure (the unique countably additive measure defined on the Borel

*σ-algebra of*R^{n}and such that the measure of a multi-rectangle is the product of the lengths of
the sides), in which case one writes d*µ = % dV , where dV denotes the Lebesgue volume element*
inR^{n}.

In probability theory, the integral of F with respect to a probability measure *µ over* Ω ^{is}
referred to as the expectation or mean value of the random “variable” F, and is denoted by

〈F〉* _{µ}*=E

_{µ}_{(F) :=}

Z

ΩF d*µ .*
The above formula implies for example that

*χ*A

®

*µ**= µ(A).*

**Exercise 2.1. Let A = [0,1] ∩ Q be the set of rationals in [0,1]; then the Lebesgue measure**
V (A) of A is zero. Moreover, the Dirichlet function D(x) - defined on [0, 1] as D(x) = 1 if x is
irrational and D(x) = 0 otherwise - is not Riemann integrable but is integrable with respect
to the Lebesgue measure over [0, 1], the value of the integral being exactly one. Indeed, since
A is countable, it can be covered by a sequence of intervals {I_{j}}_{j∈N} such that I_{j} is centered at
x_{j}∈ A and V (Ij*) = ε/2*^{j+1}, where*ε is arbitrarily small. Then, since A ⊂ S*jI_{j}, it followsV (A) ≤
V (S

jI_{j}) ≤P

j≥0V (I_{j}*) = ε, and the arbitrariness of ε implies V (A) = 0. For what concerns the*
Dirichlet function, observe that*D(x) = χ*A^{c}, so that*R D(x)dV = R χ*A^{c}*dV = µ(A*^{c}*) = 1 − µ(A) = 1.*

A good reference for probability theory is [20]. Abstract measure and integration theory is extensively treated in the analysis monograph [34].