**4.3** **The Poincaré theorem on the nonexistence of first ** **in-tegrals**

Another theorem due to Poincaré states that, under the same hypotheses of non-degeneracy
of the integrable part, genericness of the perturbation (to leading order), and under strong
regularity assumptions, the perturbed system admits only trivial first integrals, i.e. the
func-tions of the Hamiltonian itself. In what follows, for the sake of convenience, the notation is
slightly changed and the integrable part of the Hamiltonian is denoted by H_{0}(J), whereas the
perturbation terms are denoted by H_{j}(*θ, J), j ≥ 1.*

**Theorem 4.3. Let H(***θ, J;ε) = H*0*(J) + εH*1(*θ, J) + ε*^{2}. . . be analytic in all its arguments, with
H_{0}non degenerate and H_{1} generic in B. Then, no first integral F(*θ, J;ε) exists, analytic in all*
its arguments and independent ofH.

C PROOF. The scheme of the proof is the following:

1. one proves that F_{0}*:= F(θ, J;0) is independent of θ;*

2. one proves that F_{0}=F0(H_{0});

3. one proves that if F is independent of H then there exists another first integral whose zero order term is independent of H0, contrary to what proven in the previous point.

The first integral F =P

j≥0*ε*^{j}F_{j} can be formally constructed by imposing that its Poisson
bracket with H =P

j≥0*ε*^{j}H_{j} be zero. The condition
0 = {F, H} = X

j,*`≥0**ε** ^{j+`}*{F

_{j}, H

*} =X*

_{`}s≥0

*ε*^{s}
Ã

X

*j+`=s*

{F_{j}, H* _{`}*}

!

is equivalent to the hierarchy of conditions X

j,*`≥0*
*j+`=s*

{F_{j}, H* _{`}*} = 0 , s = 0,1,2... , (4.25)

to be solved one after the other.

1. The first of conditions (4.25) is

{F_{0}, H_{0}} =*∂F*0

*∂θ* ·*∂H*0

*∂J* *= ω(J) ·∂F*0

*∂θ* = 0 ,
or, passing to the Fourier coefficients

*ik · ω(J) b*F_{0,k}(J) = 0 , ∀k ∈ Z^{n} , ∀J ∈ B .

The above equation is obviously satisfied for k = 0, for any J ∈ B. For non resonant
actions, i.e. J ∈ B \S_{n−1}

r=1B_{r}*, k · ω(J) 6= 0, so that b*F_{0,k}(J) = 0 for any k ∈ Z^{n}\ {0} on a dense,
full measure set of actions. By requiring the continuity of the Fourier coefficients with

respect to the actions one immediately getsFb_{0,k}(J) = 0 for any k ∈ Z^{n}\ {0} and any J ∈ B.

One is thus left with F_{0}(*θ, J) =*Fb_{0,0}(J) := F0(J), i.e. F_{0}is independent of the angles.

An alternative argument is the following. The condition {F_{0}, H_{0}} = 0 means that one is
looking for a first integral of H_{0}, i.e. a function F_{0} which is invariant with respect to the
flow of H_{0}, so that F_{0}(*θ, J) = F*0(*θ + ω(J)t, J). For non resonant frequencies, i.e. for non*
resonant actions, the flow*θ 7→ θ +ω(J)t is ergodic on T*^{n}. As a consequence F_{0}= 〈F0〉, but
F_{0}= F0 by invariance of F_{0}, so that F_{0}(*θ, J) = 〈F*0〉 (J) for any*θ ∈ T*^{n}, any J in the dense
set of non resonant actions and, by continuity, any J ∈ B.

2. The second of conditions (4.25) reads
{F_{0}, H1} + {F1, H0} = −*∂F*0

*∂J* ·*∂H*1

*∂θ* *+ ω(J) ·∂F*1

*∂θ* = 0 ,
or, passing to the Fourier coefficients

*k · ω(J) b*F_{1,k}(J) = k ·*∂F*0(J)

*∂J* Hb_{1,k}(J), (4.26)

which must hold for any integer vector k and any action J in B. In particular, for any
given J in B_{n−1}*= ω*^{−1}(Ω_{n−1}*∩ ω(B)) there exist n − 1 linearly independent integer vectors*
k^{(1)}, . . . , k^{(n−1)} such that k^{( j)}*· ω(J) = 0 ( j = 1, . . . , n − 1). On the other hand, by the *
generic-ness of H_{1}there exist other n − 1 integer vectors q^{(1)}, . . . , q^{(n−1)}, with each q^{( j)} parallel to
k^{( j)}, such thatHb_{1,q}( j)(J) 6= 0. Since obviously q^{( j)}*·ω(J) = 0 for any j = 1, . . . , n −1, equation*
(4.26) implies q^{( j)}*· ∂F*0/*∂J = 0. Thus, for any completely resonant J there exist n − 1 *
lin-early independent vectors that are orthogonal to*ω(J) = ∂H*0/*∂J and, as a consequence,*
to*∂F*0/*∂J. Thus, for any such completely resonant J the gradients of F*0 and of H0 are
parallel to each other, i.e. there exists a function*λ(J) such that*

*∂F*0

*∂J* *= λ(J)∂H*0

*∂J* , ∀J ∈ B_{n−1}.

The latter condition can be equivalently written as dF_{0}*= λdH*0, so that*λ(J) = dF*0/dH_{0},
which implies F0(J) =F0(H0(J)) (*λ =*F_{0}^{0}) for any J ∈ Bn−1 and, by regularity, for any
J ∈ B.

3. Going back to equation (4.26), for non resonant actions and k 6= 0 one gets bF_{1,k} =
F_{0}^{0}^{(H}0)Hb_{1,k}, which extends by regularity to the whole set B. This implies

F_{1}(*θ, J) =*F_{0}^{0}^{(H}0(J))H_{1}(*θ, J) +*F1(J),

where 〈F〉1 is the undetermined mean value of F_{1} on T^{n}. The latter expression of F_{1}
yields

F_{0}*+ εF*1*+ ε*^{2}. . . = F0(H_{0}*) + ε*F_{0}^{0}^{(H}0)H_{1}*+ ε 〈F〉*1*+ ε*^{2}· · · =

= F0(H_{0}*+ εH*1*+ ε*^{2}*. . . ) + ε〈F〉*1*+ ε*^{2}. . . . (4.27)
On the other hand, by studying the third equation of the hierarchy (4.25), namely

{F_{0}, H_{2}} + {F1, H_{1}} + {F2, H_{0}} = 0 (4.28)

4.3. THE POINCARÉ THEOREM ON THE NONEXISTENCE OF FIRST INTEGRALS 75

with the same reasonings made above, one finds that

〈F〉1=F1(H0), (4.29)

F_{2}=F0^{0}(H_{0})H_{2}+1

2F0^{00}(H_{0})H_{1}^{2}+F1^{0}(H_{0})H_{1}+ 〈F〉2 , (4.30)
where 〈F〉2is undetermined. Equations (4.29) and (4.30) imply

F_{0}*+ εF*1*+ εF*2*+ ε*^{3}. . . = F0(H_{0}*+ εH*1*+ ε*^{2}H_{2}*+ ε*^{3}. . . ) +

*+ ε*F1(H_{0}*+ εH*1*+ ε*^{2}*. . . ) + ε*^{2}〈F〉2*+ ε*^{3}. . . ,

which suggests that one is actually building up a function of H. In order to prove this, suppose that

F^{(0)}:=F0(H_{0}*) + εF*1*+ ε*^{2}F_{2}*+ ε*^{3}. . .

is a first integral of the perturbed system independent of H. Let us prove that starting
from F^{(0)} one can construct another first integral such that its term of order zero (in*ε),*
denoted as usual by the subscript 0, is independent of H0. Obviously

F^{(1)}:=F^{(0)}−F0(H)

*ε* ,

as a linear combination of first integrals, is a first integral too. Now, if F_{0}^{(1)}is independent
of H_{0}one stops, otherwise one defines F_{0}^{(1)}:=F1(H_{0}) and goes on defining

F^{(2)}:=F^{(1)}−F1(H)

*ε* ,

which is still a first integral. Again, either F_{0}^{(2)} is independent of H_{0} or one goes on this
way by defining F_{0}^{(2)} :=F2(H_{0}). Such a procedure must stop after a finite number m
of steps, yielding a first integral F^{(m)} whose zero order term F_{0}^{(m)} is independent of H0.
Indeed, if this were not the case, one would have

F^{(0)} = F0*(H) + εF*^{(1)}=F0*(H) + ε*F1*(H) + ε*^{2}F^{(2)}+

= F0*(H) + ε*F1*(H) + ε*^{2}F2*(H) + ε*^{3}F^{(3)}+ . . . =X

j≥0

*ε*^{j}Fj(H),

i.e. F^{(0)} would be a function of H, against the hypothesis. Thus a first integral F^{(m)}such
that F_{0}^{(m)}is independent of H_{0}exists, but this is absurd, since in the previous point 2. it
was proven that the zero order term of any first integral must necessarily be a function
of H0. The conclusion is that any first integral F must be a function of H.B

**Exercise 4.2. Study equation (4.28) and get equations (4.29) and (4.30).**

**Appendix A**

**Hierarchy of structures in topology and** **their utility**

**Definition A.1. A metric space is a pair (X**, d) where X is a set and d : X ×X → R+is a distance,
i.e. a nonnegative function which satisfies

1. d(x, y) = 0 iff x = y (nondegeneracy);

2. d(x, y) = d(y, x) ∀x, y ∈ X (symmetry);

3. d(x, z) ≤ d(x, y) + d(y, z) ∀x, y, z ∈ X (triangle inequality).

In metric spaces one can define limits of functions and thus continuity. Indeed, given two
metric spaces (X, d) and (X^{0}, d^{0}), one says that a function f : X → X^{0}has (finite) limit*` ∈ X*^{0} as
x tends to x_{0}, and writes lim_{x→x}_{0}*f (x) = `, if for any ² > 0 there exists a δ**²*> 0 such that for all
x ∈ X \ {x0} satisfying d(x, x_{0}*) < δ** _{²}*one has d

^{0}( f (x),

*`) < ². The definition of continuous function*is obtained by setting

*` = f (x*0).

Notice that a metric space is not necessarily a linear (or vector) space. On the other hand,
if one wants to build up differential calculus, one needs the linear structure. Indeed, the
following definition of (weak, or Gateaux) differential of a function f : X → X^{0} in x_{0}∈ X with
increment h ∈ X is quite natural:

d f (x_{0}; h) := lim

t→0

1

t[ f (x0+ th) − f (x0)] .

Observe that one needs the linear combination x_{0}+ th ∈ X and the differential d f (x0; h) ∈ X^{0},
the convergence of the above limit being meant in the metric d^{0} of X^{0}. Actually, one needs
to measure the size of the objects in X and X^{0}, in order to have some control on the above
formula (for example, one would like to have that d f is “small” when h is “small”). This is
achieved when the distances d on the linear spaces X is induced by a norm k · k, such that
d(x, y) = kx − yk.

**Definition A.2. A normed space is a pair (X**, k · k) where X is a linear space and k · k : X → R_{+}
is a norm, i.e. a nonnegative function satisfying

1. kxk = 0 iff x = 0 (nondegeneracy);

77

*2. kλxk = |λ|kxk ∀λ ∈ R and ∀x ∈ X (homogeneity of degree one);*

3. kx − zk ≤ kx − yk + ky − zk (triangle inequality).

One easily checks that a normed linear space is a metric space with the distance

d(x, y) := kx − yk . (A.1)

Differential calculus is performed in normed spaces, but in order to set up all the machinery
of the differential equations one needs a stronger property. More precisely, in order to have
the uniqueness of the solution to a given differential equation one needs the completeness of
the space, namely that every Cauchy sequence converges (recall that a sequence {x_{n}}_{n∈N}⊂ X
*is a Cauchy sequence if ∀² > 0 there exist N** _{²}*∈ N such that for any pair n, m ∈ N satisfying
n ≥ m > N

*²*one has d(xn, xm

*) < ²).*

**Definition A.3. A complete normed space is called Banach space.**

Almost all the classical results in the theory of differential equations is (existence, unique-ness and regularity) hold in Banach spaces. Completeunique-ness enters the game in the proof of the Picard existence and uniqueness theorem, as follows. We recall that a differential equation

˙x(t) = u(x(t)), with initial condition x(0) = x0, is naturally solved by iterating the map P : X → X defined by

x^{0}(t) = P(x) := x0+
Z _{t}

0

u(x(s)) ds ,

with initial point x(s) ≡ x0. Now, P acts on the space B(I, X ) of bounded functions t 7→ x(t),
defined on a suitable real interval I, with value in the Banach space X . Such a space, endowed
with the metric d(x, y) := sup_{t∈I}kx(t)− y(t)k, k·k being the norm on X , turns out to be complete.

A local Lipschitz condition on u guarantees (and is optimal) that P : B → B is a contraction,
i.e. that there exists*ρ ∈]0,1[ such that d(P(x), P(y)) ≤ ρd(x, y) for any x, y close enough to x*0.
Since a contraction in a complete metric space admits a unique fixed point, then there exists a
unique ˆx(t) ∈ B(I, X ) such that ˆx(t) = P( ˆx(t)), i.e. ˆx(t) is the local solution of the given differential
equation.

It may happen that the linear space where the problem at hand is set up possesses a Eu-clidean structure, i.e. a scalar (or internal) product between its elements. When this happens it is very useful, since then concepts like angle, direction and projection become meaningful.

**Definition A.4. A Euclidean space is a pair (X**, 〈,〉) where X is a linear space and 〈,〉 : X ×X → R
is a scalar product, i.e. a real function satisfying

1. 〈x, x〉 > 0 ∀x ∈ X \ {0}, 〈0,0〉 = 0 (nondegeneracy);

2. 〈x, y〉 = 〈y, x〉 ∀x, y ∈ X (symmetry);

3.

*λx + µy, z® = λ〈x, z〉 + µ〈y, z〉 ∀λ,µ ∈ R and ∀x, y, z ∈ X (linearity in the first entry).*

79 One easily checks that a Euclidean space is a normed (and thus a metric, see (A.1)) space with the norm

kxk :=p

〈x, x〉 . (A.2)

Two vectors x, y of (X , 〈,〉) are said to be mutually orthogonal if 〈x, y〉 = 0. In a Euclidean space,
for example, the Pythagoras theorem holds, namely kx + yk^{2}= kxk^{2}+ kyk^{2} for all pairs x, y of
mutually orthogonal vectors.

Of course, the Euclidean structure is very useful in dealing with differential equations, whose appropriate environment, as explained above, is the Banach space. This justifies the introduction of a stronger topological structure, namely that of a Euclidean space which is complete with respect to the norm (A.2) naturally induced by the scalar product, which one can shortly refer as to a Euclidean-Banach space.

**Definition A.5. A Euclidean-Banach space is called a Hilbert space.**

**Remark A.1. Wherever needed, the linear structure of the space X can be considered on the**
complex fieldC, which only requires the symmetry property 2. of the scalar product to change
as follows: 〈x, y〉 = 〈y, x〉^{∗}, a star denoting complex conjugation.

**Example A.1. Quantum mechanics is the theory of certain unitary evolutions in Hilbert spaces,**
described by equations of the form

u = −i ˆ˙ Hu

where ˆH**: H → H is a real self-adjoint operator on an infinite-dimensional Hilbert space H. The**
formal solution of the above equation reads u(t) = e^{−i ˆ}^{H}^{t}u(0). The meaning of such a solution
**is completely specified in terms of the basis of H formed by the eigenvectors of ˆ**H, satisfying
Hˆ_{ϕ}_{ω}_{= E}_{ω}_{ϕ}* _{ω}*, where the set of the eigenvalues {E

*} contains, in general, both a discrete and a continuous component. If one writesu(t) =P*

_{ω}*ω*c* _{ω}*(t)

*ϕ*

*, thenu(t) =P*

_{ω}*ω*c* _{ω}*(0)e

^{−iE}

^{ω}^{t}

*ϕ*

*.*

_{ω}**Example A.2. The hydrodynamic inviscid Burgers-Hopf equation**

v_{t}+ vvx= 0 ,

where v(t, ·) is defined on X = R or X = R/Z, is a highly nontrivial example of nonlinear
Hamil-tonian PDE. A HamilHamil-tonian is H(v) = −^{1}_{6}R v^{3}dx with Poisson tensor *∂*x, so that v_{t}*= ∂*x*δH/δv.*

One easily checks that the flow of the above equation preserves all the L_{p}(X) norms k · kp :=

¡R

X| · |^{p}dx¢1/p

, for any p ≥ 1.

**Appendix B**

**Fourier series expansion of functions on** **the torus**

Let us consider a smooth function f :T^{n}→ R and its Fourier series expansion
f (*θ) =* X

k∈Z^{n}

fˆ_{k}e* ^{ik·θ}* , (B.1)

with Fourier coefficients ˆf_{k}given by
fˆ_{k}= 1

(2*π)*^{n}
Z

T^{n}e* ^{−ik·θ}*f (

*θ) d*

^{n}

*θ .*(B.2)

The larger function space where the above formal relations become meaningful isL2(T^{n},*µ*T)
[34]. Here we limit ourselves to prove the following statements linking the decay of the Fourier
coefficients of the function f to its regularity. In the sequel we make use of theR^{n}-norms

*|ξ|*s:=

C_{s,n} being a numerical constant independent of f . Conversely, suppose that three positive
con-stants A_{s}, s and R exist, such that the bounded coefficients ˆf_{k} satisfy | ˆfk| ≤ As/|k|^{s}_{1} for any

|k|1> R; then the Fourier seriesP

kfˆ_{k}e* ^{ik·θ}* is absolutely convergent, uniformly onT

^{n}, if s > n.

C PROOF. From (4.14) it follows that
(−ikj)^{s}fˆ_{k} = 1

Taking the modulus of both sides in the above identity yields

Summing the latter inequalities over j one gets

|k|^{s}_{s}| ˆfk| ≤ max

This differs from (B.3) in the presence of the norm |k|s in place of |k|1. The inequality
(B.3) is obtained by recalling that all norms are equivalent in R^{n}, so that a constant
C_{s,n}> 0 exists, such that C^{1/s}_{s,n}*|ξ|*1*≤ |ξ|*s*∀ξ ∈ R*^{n}.

On the other hand, suppose that three positive constants A_{s}, s and R exist, such that the
bounded sequence { ˆf_{k}}_{k∈Z}^{n} satisfies | ˆfk| ≤ As/|k|^{s}_{1}∀|k|1> R. Then

where C is a positive constant bounding the finite sum over |k|1≤ R, and the (optimal) estimate

has been used in the last step. Obviously, the generalized harmonic series appearing in the last row of (B.4) converges if s − n > 0.B

**Remark B.1. Under suitable hypotheses, the above result can be improved: actually the Fourier**
coefficients of absolutely continuous (a bit more than uniformly continuous) functions decay as
1/|k|1.

**Remark B.2. A large torus dimension n does not allow to preserve the same degree of regularity**
in passing from the function to the coefficient and back. For example, if | ˆfk| ≤ As/|k|^{n+1}_{1} for any

|k|1> R, then the Fourier series of f is absolutely and uniformly convergent on T^{n}, but the series
of ∇f is not.

83
**Exercise B.1. Prove the estimate (B.5). Hint: first, check that**Nn(0) = 1, N1(s) = 2,N2(s) = 4s;

secondly, prove that

Nn(s) =Nn−1(s) + 2[Nn−1(s − 1) + ··· +Nn−1(1) +Nn−1(0)] ; finally, use induction forn ≥ 3.

**Lemma B.2. If a real number** *ρ > 0 exists such that f is analytic in the complex extension of*
the torusT_{ρ}^{n}:= {z ∈ C^{n}: Re(z_{j}) ∈ T ; |Im(zj*)| ≤ ρ ; j = 1,..., n}, with M** _{ρ}*:= max

_{T}

_{ρ}^{n}| f (z)|, then

| ˆfk| ≤ M* _{ρ}*e

^{−ρ|k|}^{1}, ∀k ∈ Z

^{n}. (B.6) Conversely, suppose that two positive constantsB and

*ρ exist, such that the coefficients ˆf*ksatisfy

| ˆfk| ≤ Be^{−ρ|k|}^{1} ∀k ∈ Z^{n}. Then the function f (z) :=P

kfˆ_{k}e^{ik·z} is analytic in T^{n}* _{ρ−δ}*, with the upper
boundM

_{ρ−δ}*≤ B/(tanh(δ/2))*

^{n}

*, for any 0 < δ < ρ.*

C PROOF. Consider first the case n = 1, and define the piecewise linear, counter-clockwise oriented closed path in the complex plane

C^{+}*:= {0 ≤ x ≤ 2π; y = 0} ∪ {x = 2π;0 ≤ y ≤ ρ} ∪ {0 ≤ x ≤ 2π; y = ρ} ∪ {x = 0;0 ≤ y ≤ ρ} .*
Let us consider also the clockwise oriented, closed pathC^{−}, reflected ofC^{+} with respect
to the x-axis. Since e^{−ikz}f (z) is analytic in an open set containingC^{+}^{, one has}

0 = I

C^{+}e^{−ikz}f (z) dz =
Z 2*π*

0

e^{−ikx}f (x) dx +
Z 0

2*π*e^{−ik(x+iρ)}*f (x + iρ) dx +*
+

Z _{ρ}

0

e*−ik(2π+i y)*f (2*π + i y) idy +*
Z _{0}

*ρ* e^{−ik(0+i y)}f (0 + i y) idy .

The two integrals in the second row above cancel out each other by periodicity of f along the x direction, so that

fˆ_{k}= 1
2*π*

Z _{2}_{π}

0

e^{−ikx}f (x) dx = e^{k}* ^{ρ}*
2

*π*

Z _{2}_{π}

0

e^{−ikx}*f (x + iρ) dx .*

Now, if k = −|k| ≤ 0, the latter relation yields the searched estimate

| ˆfk| ≤ M*ρ*e* ^{−|k|ρ}* .

In the case k ≥ 0 the same result is reached by choosing C^{−} as the starting contour
of integration. The generalization to the case n > 1 is almost trivial. One defines the
contourC_{j}^{±} relative to the variable zj= xj+ i yj as above ( j = 1,..., n). The analyticity of
e^{−ik·z}f (z) with respect to z_{j}implies

I

C_{j}^{±}e^{−i(k}^{1}^{z}^{1}^{+···k}^{n}^{z}^{n}^{)}f (z_{1}, . . . , z_{n}) dz_{j}= 0 ∀ j = 1, . . . , n .

By integrating e^{−ik·z}f (z) with respect to each variable z_{j} on the contour C_{j}^{σ}^{j}^{, where}
*σ*j= −sign(kj), implies

fˆ_{k}=e^{−ρ(|k}^{1}^{|+···+|k}^{n}^{|)}
(2*π)*^{n}

Z

T^{n}e^{−i(k}^{1}^{x}^{1}^{+···k}^{n}^{x}^{n}^{)}f (x_{1}*+ iσ*1y_{1}, . . . , x_{n}*+ iσ*ny_{n}) dx_{1}· · · dxn.
The latter relation provides then the estimate (B.6).

Suppose now, conversely, that two positive constants B and *ρ exist, such that the *
se-quence { ˆf_{k}}_{k∈Z}^{n} satisfies | ˆfk| ≤ Be^{−ρ|k|}^{1} ∀k ∈ Z^{n}. One has

¯

¯

¯

¯

¯ X

k∈Z^{n}

fˆ_{k}e^{ik·z}

¯

¯

¯

¯

¯

≤ X

k∈Z^{n}

Be^{|k}^{1}^{|(|y}^{1}*|−ρ)+···+|k*n|(|yn*|−ρ)*.

Notice that the series on the right hand side requires |yj*| < ρ for any j = 1, . . . , n in order*
to converge. Setting |yj*| = ρ − δ one gets the estimate*

¯

¯

¯

¯

¯ X

k∈Z^{n}

fˆ_{k}e^{ik·z}

¯

¯

¯

¯

¯

≤ B X

k∈Z^{n}

e^{−δ|k|}^{1}= B
Ã

X

j∈Z

e^{−δ j}

!n

= B

(tanh(*δ/2))*^{n} ,
which proves the upper bound on the maximum M* _{ρ−δ}* of f (z) :=P

kfˆ_{k}e^{ik·z} in T_{ρ}^{n}. The
analyticity of f (z) in the same domain is an obvious consequence of the exponential decay
of its Fourier coefficients, which in turn implies that any partial derivative of f , of any
order, admits a Fourier series expansion absolutely and uniformly convergent onT^{n}* _{ρ}*.B

**Bibliography**

[1] V. I. Arnol’d, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1989.

[2] V. I. Arnol’d, Ordinary Differential Equations, MIT Press, 1978.

[3] V. I. Arnol’d, Metodi geometrici della teoria delle equazioni differenziali ordinarie, Editori Riuniti, 1989.

[4] V. I. Arnol’d, Lectures on Partial Differential Equations, Springer, 2004.

[5] V. I. Arnol’d and A. Avez, Ergodic problems of classical mechanics, W. A. Benjamin, 1968.

[6] V. I. Arnol’d, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, 3rd edition, Springer, 2006.

[7] J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, 1987.

[8] G. Boffetta and A. Vulpiani, Probabilità in Fisica, Springer, 2012.

[9] N. N. Bogolyubov, Y. A. Mitropolsky: Asymptotic Methods in the Theory of Non-Linear Oscillations, New York, Gordon and Breach, 1961.

[10] S. Bonometto, Cosmologia & cosmologie, Zanichelli, 2008.

[11] S. G. Brush, The Kinetic Theory of Gasses, Imperial College Press, 2003.

**[12] R. Clausius, On a Mechanical Theorem Applicable to Heat, Philosopical Magazine 40**
(1870), 122-127.

[13] G. W. Collins, The Virial Theorem in Stellar Astrophysics, Pachart Pub-lishing House, 1978; available for free on the NASA ADS archive:

http://ads.harvard.edu/books/1978vtsa.book/

[14] A. Fasano and S. Marmi, Analytical Mechanics, Oxford University Press, 2006.

[15] J. N. Franklin, Matrix Theory, Dover, 1968.

[16] F. R. Gantmacher, Lezioni di Meccanica Analitica, Editori Riuniti, 1980.

[17] J. W. Gibbs, Elementary Principles in Statistical Mechanics, Dover, 1960 (originally pub-lished by the Yale University press, 1902).

85

[18] P. R. Halmos, Lectures on Ergodic Theory, Chelsea Publishing Company, 1956.

[19] K. Huang, Statistical Mechanics, John Wiley & Sons, 1987.

[20] J. Jacod and P. Protter, Probability Essentials, Springer-Verlag 2004.

[21] M. Kac, Probability and related topics in physical sciences, Interscience Publishers, 1959.

[22] A. I. Khinchin, Mathematical Foundations of Statistical Mechanics, Dover, N.Y., 1949.

[23] A. N. Kolmogorov and S. V. Fomin, Elementi di teoria delle funzioni e di analisi funzionale, Mir, 1980.

**[24] V. V. Kozlov, An extension of the Hamilton-Jacobi method, J. Appl. Maths. Mechs. 60**
(1996), 911-920.

[25] L. D. Landau and E. M. Lifšits, Meccanica, Fisica Teorica 1, Editori Riuniti, 1991.

[26] L. Markus and K. R. Meyer, Generic Hamiltonian Dynamical Systems are neither
**Inte-grable nor Ergodic, Memoirs of the AMS 144, 1974.**

[27] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd edition, Springer, 1999.

[28] V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Dover, 1989.

[29] W. Pauli, Thermodynamics and the Kinetic Theory of Gases, Pauli Lectures on Physics 3, Dover, 2000.

**[30] H. Pollard, A Sharp Form of the Virial Theorem, Bull. Amer. Math. Soc. 70 (1964), **
703-705.

[31] H. Pollard, Mathematical Introduction to Celestial Mechanics, Prentice-Hall, 1966.

[32] G. Prodi, Analisi Matematica, Bollati Boringhieri, 1970.

[33] G. Prodi, Lezioni di Analisi Matematica 2, Bollati Boringhieri, 2011.

[34] W. Rudin, Real and Complex Analysis, 3-rd ed., McGraw-Hill, 1987.

[35] L. Salce, Lezioni sulle matrici, Decibel-Zanichelli, 1993.

[36] D. G. Saari, Collisions, Rings, and Other Newtonian N-Body Problems, AMS, 2005.

[37] J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical systems, Springer-Verlag, 2007.

[38] G. E. Shilov, Elementary Functional Analysis, Dover, N.Y., 1996.

[39] C. J. Thompson, Mathematical Statistical Mechanics, Macmillan, 1971.

BIBLIOGRAPHY 87 [40] G. E. Uhlenbeck, G. W. Ford and E. W. Montroll, Lectures in Statistical Mechanics,

Amer-ican Mathematical Society, 1963.

[41] F. Zwicky, On the Masses of Nebulae and of Clusters of Nebulae, The Astrophysical
**Jour-nal, 86 (1937), 217-246.**