Before entering the discussion we should make an agreement on the meaning of the term integrable system

INTEGRABLE SYSTEMS

The present chapter deals with general methods that should allow us to integrate the Hamilton’s equations, using the tools developed in the previous chapters. This was indeed the dream of the great mathematicians, after Newton, till the end of the XIX century. Nowadays we are well aware that integrable systems are rather exceptional, and this is indeed a good justification of the fact that most textbooks contain the same examples — just a few which are the classical and more interesting ones. Nevertheless, integrable systems represent an excellent first approximation of interesting mechanical systems, and are the starting point of classical perturbation theory. It is not far from reality to say that in most cases the goal of perturbation methods is to reduce a system of differential equations to a form as close as possible to an integrable one.

Before entering the discussion we should make an agreement on the meaning of the term integrable system. In view of the theorem of existence and uniqueness of the solutions of a system of differential equations every Hamiltonian system can be said to be integrable provided some mild regularity conditions are satisfied by the Hamil- tonian function. This is useful, of course, if one is interested in computing the orbit corresponding to a given initial datum, e.g., with numerical methods.¹ However, we should keep in mind that the theorem has a local character: in our case it assures only the existence of the solution for some time interval. The process of continuation of a given solution may be used in order to establish the existence of the solution for larger time intervals, but it gives essentially no information about the global behaviour of the orbits. In the framework of Hamiltonian systems (although this is not a true restric-

1 The most common numerical methods for solving differential equations are indeed based on the possibility of writing the first few terms of the Taylor expansion of the solution.

The computation of the orbit is performed by repeating an elementary iteration step:

starting from the initial point at time 0 one computes the (approximate) point at time τ; then the new point is used as initial point for the next step, and so on. However, the expansion is only local. Moreover, small error which are unavoidably introduced at every step may accumulate. Proving that the computed orbit remains close to the true orbit for a long time interval is, generally speaking, an hard problem.

tion) it is customary to assign a more restricted meaning to the word integrability. In some sense, one asks for being able to write the solution for all times.

The traditional interpretation involves the concept of integrability by quadratures.

This means that: the solution should be written in terms finite number of algebraic operations, including inversion of functions, and of computation of integrals of known functions (quadrature). The integration method discussed in sect. 1.3 for systems with one degree of freedom is a good example. In the framework of Hamiltonian theory Liouville’s theorem can be considered as the most advanced general result in this direction. The paradigm model is represented by an Hamiltonian depending only on the momenta p₁, . . . , p_n, i.e., H = H(p₁, . . . , p_n) , which is trivially integrable.

In short, Liouville’s theorem says that if a Hamiltonian system possesses enough first integrals then the Hamiltonian can be given the form above with a suitable coordinate transformation.

In more recent times more attention is paid to the global description of the behav- ior of the solutions, with particular attention to the existence of periods, or frequen- cies. Thus, most authors impose the sharper condition that the coordinates q₁, . . . , q_n conjugated to p₁, . . . , p_n are actually angles, that is, q ∈ Tⁿ. In the latter case the canonically conjugated variables p, q are called action–angle variables. This seems to be a very strong condition: for instance, the problem of a mass point freely moving on the space can not be described by action–angle variables in strict sense because there are no periods. However, such a strong attitude can be justified a posteriori. Indeed, small perturbations of an integrable system that admits action–angle variables typ- ically produce a very complicated dynamical behavior, which is still not completely understood.²

In view of this discussion, we consider the following problem: Assume we are given a Liouville–integrable system. Can we introduce action–angle variables? The answer to this question is provided by the theorem of Arnold–Jost.³

The theorems of Liouville and of Arnold–Jost constitute the main contents of this chapter. A general discussion of the dynamical behaviour of an integrable system is also included.

2 A typical situation arising in Mechanics is the study of a system of many particles, that may be either free of moving all around the space under the mutual interactions, as is the case of our planetary system or of an atomic system, or may be subjected to some constraints, as, e.g., in the case of a rigid body. The first step usually consists in exploiting the conservation of the total momentum by eliminating the motion of the center of mass, which is a trivial one being that of a free particle. Then in many cases action–angle variables may be introduced in some approximation.

3 The use of action–angle variables was well known in connection with classical problems like the planetary motions and the motion of a rigid body. During the first decades of this century it has become relevant also in connection with the first developments of quantum theory. A classical and valuable reference is M. Born’s treatise [18]. The theorem of Arnold–Jost states that integrability in classical mechanics is strongly connected with the existence of action–angle variables.

3.1 Involution systems

The investigation of integrability is mainly based on the existence of independent first integrals. In the framework of Hamiltonian systems a relevant role is played by first integrals with vanishing mutual Poisson bracket.

A system of r functions {Φ1(q, p), . . . , Φ_r(q, p)} is said to be an involution system if the functions are independent, i.e.,

rank

 ∂(Φ₁, . . . , Φ_r)

∂(q₁, . . . , q_n, p₁, . . . , p_n)



= r ,

and the Poisson bracket between any two functions vanishes, i.e., {Φj, Φ_k} = 0 for j, k = 1, . . . , r.

3.1.1 Some geometrical properties.

We prove some lemmas that will be used in the rest of the chapter.

Lemma 3.1: An involution system contains at most n independent functions, where n is the number of degrees of freedom.

Proof. It is convenient to use the compact notation. At any point z ∈ F the sym- plectic gradients (J∂_zΦ₁, . . . , J∂_zΦ_r) span a r–dimensional subspace which is isotropic, due to the involution property satisfied by the functions. By lemma 2.5 the dimension of such a subspace can not exceed n. We conclude that r ≤ n. Q.E.D.

Example 3.1: Involution systems constructed using the canonical coordinates. The most trivial but useful example is given by the canonical coordinates themselves.

Consider any partition J, K of {1, . . . , n}; then the n functions {qj}j∈J ∪ {pk}k∈K

form an involution system.

Other examples may be easily constructed by making reference to known inte- grable systems. For instance, let the phase space be R³× R³, with canonical coordi- nates x, y, z and momenta px, py, pz. The latter three quantities are the components of the momentum, and form an involution system. This reminds us the case of a free particle in the ordinary euclidean space. On the other hand, it is just a particular case of the first example, since it corresponds to a partition which selects only the momenta.

Example 3.2:Using the angular momentum in spherical coordinates It seems spon- taneous to try to construct involution systems by using the three components of the angular momentum, namely M_x = yp_z − zpy, M_y = zp_x − xpz, M_z = xp_y − ypx. However, it is immediately seen that the latter three quantities are independent, but not in involution: this has been shown in example 1.10. Replacing some components of the angular momentum with some components of the momentum does not help, for the same reason.

An involution system may be constructed by considering one of the components of the angular momentum, for instance M_z, and the quantity Γ² = M_x² + M_y² + M_z², namely the square of the norm of the angular momentum. The latter two quantities

are indeed in involution. A third function in involution with Mx and Γ² is, e.g., E = 1

2m(p²_x+ p²_y+ p²_z) + V (r) ,

where r =px² + y²+ z², and V (r) an arbitrary (differentiable) function.

The same example can be reformulated using spherical coordinates. The phase space is (0, +∞) × (0, π) × T × R³, with canonical coordinates r, ϑ, ϕ, pr, pϑ, pϕ. The three functions

J = p_ϕ , Γ² = p²_ϑ+ J²

sin²ϑ , E = 1 2m



p²_r+ Γ² r²



+ V (r)

form an involution system. One will recognize here the first integrals of the problem of motion under central forces, example 1.12.

Example 3.3: Harmonic oscillators A last example is constructed by considering a system of harmonic oscillators. Let the phase space to be R²ⁿ, with canonical coordi- nates x, y. The n functions

Φ₁ = x²₁+ y²₁

2 , . . . , Φ_n = x²_n+ y_n² 2 form an involution system.

Although apparently trivial this example plays a central role in studying the small oscillations of a system in the neighbourhood of an equilibrium, since it represents a remarkable first approximation.

In the rest of the chapter we shall need the following technical

Lemma 3.2: Let Φ₁, . . . , Φ_n be an involution system on the phase space F . Then at every point P ∈ F there is a partition J, K of {1, . . . , n} such that

det ∂(Φ₁, . . . , Φ_n)

∂(qJ, pK)

 6= 0 .

The proof is a straightforward adaptation of that of lemma 2.7, and is left to the reader.

In classical treatises, and later in the present section, a wide use is made of the condition

(3.1) det ∂(Φ₁, . . . , Φ_n)

∂(p1, . . . , pn)

 6= 0 .

Such a condition may appear restrictive, in that it seems to attribute a privileged role to the momenta p. In this respect it should be remarked that the independence of the Φ’s is expressed as

rank

 ∂(Φ1, . . . , Φn)

∂(q₁, . . . , q_n, p₁, . . . , p_n)

 6= 0 ,

whis follows from the previous condition (3.1), but does not imply it. Lemma 3.2 means that in the latter n×2n Jacobian matrix it is always possible to select a n×n submatrix

with non zero determinant where derivatives with respect to both conjugate variables q_j, p_j do not appear. This is tantamount to saying that by appropriately exchanging some pairs of conjugate variables with the canonical transformation of example 2.13 we can always manage so that condition (3.1) is satisfied.

3.1.2 The Hamiltonian flow as a canonical transformation

Consider a point (q, p) of the phase space, and let (qt, pt) = φ^t(q, p) be the transformed point under the flow φ^t generated by a canonical system with Hamiltonian H(q, p).

For fixed t we can consider (qt, pt) as new coordinates of the point (q, p). That is, we consider the flow as generating a coordinate transformation on the phase space.

Lemma 3.3: Let φ^t denote the flow generated by the Hamiltonian H(q, p). Then for every fixed t the transformation (qt, pt) = φ^t(q, p) is canonical.

Proof. It is enough to prove that for every t the fundamental Poisson brackets are preserved, namely that

{qj,t, q_k,t}q,p= {pj,t, p_k,t}q,p = 0 , {qj,t, p_k,t}q,p= δ_j,k .

This is true for t = 0, because φ⁰(q, p) = (q, p) is the identity. Let us prove that at every point (q, p) of the phase space we have

d

dt{q^j,t, qk,t} = d

dt{p^j,t, pk,t} = d

dt{q^j,t, pk,t} = 0 ,

To this end, recalling again that qj,0 = qj, pj,0 = pj and using Hamilton’s equations we have

q_j,t= q_j + t∂H

∂p_j + . . . , p_j,t = p_j− t∂H

∂q_j + . . . , where the dots stand for terms of higher order in t. Thus we have

{qj,t, q_k,t} = {qj, q_k} + t

n∂H

∂pj

, q_ko +n

q_j, ∂H

∂pk

o

+ . . . , which in turn means that one has

d

dt{q^j,t, qk,t} =n∂H

∂p_j, qk

o +n

qj,∂H

∂p_k o

= − ∂

∂p_k

∂H

∂p_j + ∂

∂p_j

∂H

∂p_k = 0 . With a similar calculation we get

d

dt{p^j,t, pk,t} = −n∂H

∂q_j, pk

o

−n

pj,∂H

∂q_k o

= 0 , d

dt{q^j,t, pk,t} = n∂H

∂p_j, pk

o

−n

qj,∂H

∂q_k o

= 0 .

Since the fundamental Poisson brackets have a zero time derivative at every point,

they keep a constant value along every orbit. Q.E.D.

The proposition has a suggestive geometrical interpretation: the Hamiltonian flow can be seen as the unfolding of a canonical transformation parametrically depending on time.

3.1.3 Variational equations and first integrals

Let a system of differential equations (which needs not be Hamiltonian) (3.2) ˙x_j = X_j(x₁, . . . , x_n) , 1 ≤ j ≤ n ,

be given and let x(t) be an orbit with initial point x₀. Let also x₀+δx₀ be a point close to x₀, with an infinitesimal increment δx₀, and let x(t) + δx(t) be the corresponding orbit, so that it is a solution of the differential equations

d

dt(xj+ δxj) = Xj(x1+ δx1, . . . , xn+ δxn)

= X_j(x₁, . . . , x_n) +

n

X

l=1

∂Xj

∂x_l (x₁, . . . , x_n) δx_l+ . . . .

where the dots denote terms of higher order in δx . Since x(t) is assumed to be a solution of the equation ˙x = X(x), one immediately gets that δx(t) obeys the so called variational equation

(3.3) d

dtδxj =

n

X

l=1

∂X_j

∂xl δxl , 1 ≤ j ≤ n , where the functions ^∂X_∂x^j

l(x1, . . . , xn) must be evaluated along the known solution x(t).

A similar procedure applies to the Hamiltonian case. Let us do it in detail, recalling that the canonical equations have the rather particular form

(3.4) ˙qj = ∂H

∂p_j , ˙pj = −∂H

∂q_j , 1 ≤ j ≤ n .

Let us denote by δq_j, δp_j respectively the increments with respect to the variables qj, pj. Then the variational equations are

(3.5)

d dtδq_j =

n

X

l=1

 ∂²H

∂pj∂ql

δq_l+ ∂²H

∂pj∂pl

δp_l



d

dtδp_j = −

n

X

l=1

 ∂²H

∂qj∂ql

δq_l+ ∂²H

∂qj∂pl

δp_l



An interesting relation between first integrals and variational equations is given by the following

Proposition 3.4: Let Φ be a first integral of the canonical system with Hamiltonian H(q, p). Then a solution of the variational equations (3.5) is

(3.6) δq_j = τ ∂Φ

∂p_j , δp_j = −τ∂Φ

∂q_j , 1 ≤ j ≤ n ,

τ X_Φ(φ^t2z₀) . . . φ^t1z₀

τ X_Φ(φ^tz₀)

. . .

τ X_Φ(z0)

z₀

φ^t2z₀

φ^tz₀ τ X_Φ(φ^t1z₀)

φ^t1z^′

0

φ^tz^′

0

. . . φ^t2z^′

0

z^′

0

Figure 3.1. A solu- tion of the variational equation provided by a first integral.

where τ 6= 0 is an arbitrary constant.

Proof. In view of the linearity of the variational equations it is enough to prove the statement for τ = 1. By differentiating the relation {Φ, H} = 0 we immediately get

 ∂Φ

∂p_j, H

 +

 Φ,∂H

∂p_j



= 0 ,  ∂Φ

∂q_j, H

 +

 Φ,∂H

∂q_j



= 0 , 1 ≤ j ≤ n , that is

d dt

∂Φ

∂p_j = ∂H

∂p_j, Φ



, d

dt

∂Φ

∂q_j = ∂H

∂q_j, Φ

 . Writing in explicit form the r.h.s. of these equations we get

d dt

∂Φ

∂pj

=

n

X

l=1

 ∂²H

∂pj∂ql

∂Φ

∂pl − ∂²H

∂pj∂pl

∂Φ

∂ql

 , d

dt

∂Φ

∂qj =

n

X

l=1

 ∂²H

∂qj∂ql

∂Φ

∂pl − ∂²H

∂qj∂pl

∂Φ

∂ql

 ,

which in view of (3.6) coincides with (3.5). Q.E.D.

The proposition essentially says that the increment (δq, δp) in (3.6) is the linear approximation in τ of the canonical flow due to Φ(q, p), as illustrated in fig. 3.1. In order to simplify the notations denote by XΦ the canonical vector field generated by the first integral Φ and use the compact notation by denoting z = (q, p). Take z as the initial point of an orbit and and z^′ = z + τ XΦ(z) the initial point of a close orbit.

Then the statement of the theorem says that

(3.7) τ X_Φ(φ^tz) = φ^t τ X_Φ(z)
.

That is: the linear approximation of the increment at the point φ^tzis the approximated flow at time τ of the Hamiltonian field generated by Φ . This suggests a stronger property, namely that the flow of Φ sends orbits of H into orbits of H.

Figure 3.2. Illustrating the commutation of flows.

Exchanging the order of the flows one does not find the same ending point, in general.

(q, p) φ^t_F

φ^t_F(q, p) φ^τ_G

φ^τ_Gφ^t_F(q, p)

φ^τ_G

φ^t_F

φ^t_Fφ^τ_G(q, p) φ^τ_G(q, p)

3.1.4 Commutation of canonical flows

Let two functions F (q, p) and G(q, p) be given. We can consider both of them on the same foot, i.e., both generate a Hamiltonian vector field, and so a canonical flow. Now, the question is the following, (seefig. 3.2). Given an initial point z = (q, p), let us follow the flow of F up to a time t , thus getting at the point φ^t_Fz, and then follow the flow of G up to time τ , ending up in φ^τ_G◦ φ^tFz. Then exchange the order of the flow, thus moving from z to φ^τ_Gz along the flow of G and then to φ^t_F ◦ φ^τGz along the flow of F , the times t and τ being unchanged. As a general fact one cannot expect the two end points to be the same.

Let us evaluate the difference, assuming that the times t and τ are small. It is convenient to use again the compact notation denoting with z = (q, p) a point of the phase space. Recall also that the canonical equations may be written in terms of Lie derivatives as ˙z = L_Fzand L_Gz, respectively. Expand the flow of F with initial point z₀ up to second order in t, thus getting

φ^t_Fz₀ = z₀+ tL_Fz₀+ t²

2L²_Fz₀+ o(2) ,

the symbol o(2) standing for higher order terms. Apply now the flow of G at time τ , still keeping terms up to second order in t and τ . We have

φ^τ_G ◦ φ^tFz₀ = z0+ tLF + τ LG
z₀+ t²

2L²_F + τ tLGLF + τ² 2 L²_G



z₀+ o(2) . Exchanging the role of F and G we get the similar expression

φ^t_F ◦ φ^τGz₀ = z₀+ τ L_G+ tL_F
z₀+ τ²

2 L²_G+ tτ L_FL_G+ t² 2L²_F



z₀+ o(2) . Comparing the latter two expressions we get

(3.8) φ^τ_G◦ φ^tF − φ^tF ◦ φ^τG
z₀ = τ t L_GL_F − LFL_G
z₀ = τ tL_{F,G}z+ o(2) ,

the symbol o(2) standing for terms of higher order. We conclude that the difference between the final points of the two paths depends on the vector field generated by {F, G}, which is named the commutator between the vector fields of F and G.

One can also follow a different path such as, e.g., φ^(1−β)_G τ ◦ φ^(1−α)tF ◦ φ^βτG ◦ φ^αtF z

with 0 < α < 1 and 0 < α < 1, or any other more elaborated splitting in multiple steps. The result will generally depend on the path, so that different splittings will end up on different points. The interesting fact is that if F and G are in involution then the final points coincide.

The argument is based on two facts, The first one is the following obvious prop- erty of canonical transformations. Let (Q, P ) = C (q, p) be canonical, and denote by C H)(q, p) = H(Q, P )

(Q,P )=C (q,p) the transformed Hamiltonian. Pick a point (q, p) = C (Q, P ) of the phase space. The obvious fact is that the image C Ω(Q, P )
of an orbit of H(Q, P ) is an orbit of C H
(q, p). That is, the canonical map C sends orbits of H(Q, P ) into orbits of C H(q, p)
.

The second fact comes from considering the one–parameter family of canonical transformations (q_τ, p_τ) = φ^τ_Φ(q, p) generated by the canonical flow of a function Φ(q, p), parameterized by τ in some interval around zero.

Lemma 3.5: Let Φ(q, p) be a first integral of the Hamiltonian H(q, p). Then we have

(3.9) φ^τ_ΦH
(q_τ, p_τ) = H(q_τ, p_τ) .

That is: the Hamiltonian H is invariant under the canonical flow generated by Φ.

Proof. We are actually considering H as a dynamical variable under the flow of Φ at time τ . Therefore for every τ in the allowed interval we have

dH

dτ = {H, Φ} = 0 ,

in view of Φ being a first integral for H. The claim follows because for τ = 0 the

transformation is the identity. Q.E.D.

As we see, the Hamiltonian H and its first integral Φ play a symmetric role. Thus we can resume the whole discussion of the present section in the following general Proposition 3.6: Let the functions F (q, p) and G(q, p) be in involution, and con- sider the canonical flows at times t and τ generated by F and G , respectively, i.e.

(3.10) (q_t, p_t) = φ^t_F(q, p) , (q_τ, p_τ) = φ^τ_G(q, p) . Then the following statements hold true.

(i) The function G is invariant for the canonical flow generated by F ; conversely, the function F is invariant for the canonical flow generated by G . That is:

F (q_τ, p_τ) 

(qτ,pτ)=φ^τ_G(q,p) = F (q, p) , G(q_t, p_t) 

(qt,pt)=φ^t_F(q,p) = G(q, p) .

Σ₀

P₀

M₀ P

c Φ

0 Φ

α

χ χ

α

χ

φ^αP

Figure 3.3. Illustrating the local coordinates induced by the flow of the in- volution system Φ¹(q, p), . . . , Φn(q, p) . For graphical reasons the notation P⁰ for (q⁰, p⁰) and P for (q, p) is used in the figure.

(ii) The flows (3.10) do commute, i.e., for every (q, p) we have φ^τ_G ◦ φ^tF(q, p) = φ^t_F ◦ φ^τG(q, p) .

3.1.5 Complete involution systems and coordinates induced by the flow

Proposition 3.6 turns out to be very useful when we have a complete involution system Φ₁(q, p), . . . , Φ_n(q, p) on a phase space F . Indeed this enables us to introduce local coordinates constructed through the canonical flows of the functions. This is illustrated in fig. 3.3. The crucial point is the following. Let φ^α₁¹, . . . , φ^α_nⁿ be the flows of Φ₁, . . . , Φ_n up to time α₁, . . . , α_n, respectively. Apply the flows φ^α₁¹, . . . , φ^α_nⁿ to a point (q, p) ∈ F in any order: the result will always be the same, in view of the property that the flows do commute. Thus, we shall simply denote

φ^α = φ^α₁¹ ◦ . . . ◦ φ^αnⁿ

for α in some neighbourhood of the origin of Rⁿ.

Local coordinates can be constructed exploiting the existence and the commuting property of the flows, as illustrated in fig. 3.3. The claim is that the values of the functions Φ₁, . . . , Φ_n and the times α₁, . . . , α_n of the corresponding canonical flows define a local coordinate system.

The formal statement is given by⁴

Lemma 3.7: Let Φ₁(q, p), . . . , Φ_n(q, p) be a complete involution system. Pick a point (q0, p0) ∈ F and let Φ¹(q0, p0) = c1, . . . , Φn(q0, p0) = cn, with c ∈ Rⁿ. Then there exist a n-dimensional manifold Σ₀, a neigbourhood V_Φ ⊂ Rⁿ of c, a neighbourhood Vα ⊂ Rⁿ of the origin and a neighbourhood U ⊂ F of (q⁰, p0) such that the following holds true: there exists a diffeomorphism χ : V_α× VΦ → U mapping (α, Φ) ∈ Vα× VΦ

to (q, p) = χ(α, Φ) ∈ U satisfying

(3.11) χ(0, Φ) ∈ Σ0 , χ(α, Φ) = φ^αχ(0, Φ) . Proof. Let

M₀ =(q, p) ∈ F : Φ1(q, p) = c₁, . . . , Φ_n q, p) = c_n ,

so that (q₀, p₀) ∈ M0. By lemma 3.2 there is a partition J, K of {1, . . . , n} such that at the point q0, p0 we have

det ∂(Φ₁, . . . , Φ_n)

∂(q_J, p_K)

 6= 0 .

Let us assume for simplicity that det_∂Φ

j

∂pk

 6= 0 ; this may be achieved by possibly exchanging some pairs of conjugate coordinates. By the implicit function theorem the relations Φ₁(q, p) = c₁, . . . , Φ_n = c_n(q, p) can be inverted in a neighbourhood of (q0, p0) , so as to give p1 = p1(Φ, q), . . . , pn = pn(Φ, q) . Set now q = q0and let Φ ∈ V^Φ, a neighbourhood of Φ = c. Then the functions p₁ = p₁(Φ, q₀), . . . , p_n = p_n(Φ, q₀) determine a local manifold Σ0 parameterized by coordinates Φ which is transversal to the manifold M₀. Let now q, p(Φ, q)
∈ Σ0, and let M_Φ be the manifold of constant Φ through (q, p). As we have observed, for any α ∈ V^α ⊂ Rⁿ, a neighbourhood of the origin, the flow φ^α(q, p) is defined and depends only on α. Let now the map χ : V_α × VΦ → F be defined by (3.11), and let U = χ(Vα × VΦ) . In view of the smoothness and differentiability with respect to parameters of the solutions of differential equations the map is a diffeomorphism between V_α× VΦ and U . Q.E.D.

Remark. The choice of the manifold Σ₀ in the proof is the most natural one. A more general choice is

(3.12) p1 = p1(Φ, q0) + w1(Φ), . . . , pn = pn(Φ, q0) + wn(Φ)

with arbitrary (differentiable) functions w₁(Φ), . . . , w_n(Φ). This corresponds to a dif- ferent choice of the origin of the coordinates α on every manifold MΦ.

4 A similar statement is easily made for a generic n–dimensional manifold where n inde- pendent and commuting vector fields are defined. The peculiar aspect of the lemma is that the Hamiltonian structure allows us to make an effective use of the Hamiltonian vector fields generated by a complete involution system, combining both the existence of invariant surfaces and the flow along the surfaces.

3.1.6 Complete involution systems and Liouville variables

The following proposition claims that the map χ of proposition 3.7 may be constructed by quadratures, with the further property of being canonical. Hereafter the canonical ccordinates so found will be called Liouville variables.

Proposition 3.8: Let {Φ1(q, p), . . . , Φ_n(q, p)} be an involution system. Then there exists a local canonical transformation to new variables α, Φ

q = q(α, Φ) , p = p(α, Φ) . With the non restrictive hypothesis

(3.13) det ∂(Φ₁, . . . , Φ_n)

∂(p₁, . . . , p₁)

 6= 0 .

the generating function of the canonical transformation is constructed by quadrature as

(3.14) S(Φ, q) =

Z X

j

pj(Φ, q)dqj ,

where p₁(Φ, q), . . . , p_n(Φ, q) are obtained by inversion of Φ₁(q, p), . . . , Φ_n(q, p) .

Corollary 3.9: The Liouville variables of proposition 3.8 are determined up to a canonical transformation with generating function

(3.15) W (Φ, α) =X

j

Φ_jα_j + F (Φ) ,

where F (Φ) is an arbitrary function. Equivalently, one can add an arbitrary function F (Φ) to the generating function S(Φ, q) defined by (3.14).

The proof of the corollary is trivial, and is left to the reader. By going back to the remark after the proof of lemma 3.7 we realize that adding a function F (Φ) corresponds to the arbitrary choice of the surface Σ₀ corresponding to α = 0 . However, the choice of the functions w_j(α) is not arbitrary if we want the coordinates to be canonical. For the new coordinates α^′_j = α_jw_j(Φ) must preserve the fundamental Poisson brackets, and in particular

{α^′j, α^′_k} = δj,k+ {α^′j, w_k(Φ)} + {wj(Φ), α^′_k} = δj,k , which in turn gives

∂w_j

∂Φ_k − ∂w_k

∂Φ_j = 0 . We conclude that w_j(Φ) = ^{∂W (Φ)}_∂Φ

j with an unique arbitrary function.

Corollary 3.10: The coordinates α1, . . . , αn of proposition 3.8 are the times of the canonical flows generated by the first integrals Φ₁, . . . , Φ_n, respectively, up to a Φ depending translation as in corollary 3.9.

Proof. In the new Liouville variables α, Φ the canonical equations for Φk are ˙αj = δ_j,k and ˙Φ_j = 0 , with j = 1, . . . , n . Thus α_k is the time of the flow generated by Φ_k,

for k = 1, . . . , n , as claimed. Q.E.D.

Proof of proposition 3.8. In view of lemma 3.2 condition (3.13) is not restrictive, since it can always be fulfilled by exchanging some of the coordinates with the conju- gated momenta. In view of that condition, we can invert the functions Φ₁, . . . , Φ_nwith respect to p₁, . . . , p_n, thus getting p₁ = p₁(Φ, q), . . . , p_n = p_n(Φ, q). Consider the dif- ferential formP

jp_j(Φ, q)dq_j; we prove that it is exact. To this end, let us differentiate the identity

Φj = Φj(q, p)

p=p(Φ,q)

with respect to Φ, q, namely taking into account that in the r.h.s. p must be replaced by its expression in terms of Φ, q. This gives

dΦ_j =X

k,l

∂Φ_j

∂pk

 ∂p_k

∂Φl

dΦ_l+ ∂p_k

∂ql

dq_l



+X

l

∂Φ_j

∂ql

dq_l .

By comparison of the coefficients of dq, dΦ we get the identities X

k

∂Φ_j

∂pk

∂p_k

∂Φl

= δ_j,l , X

k

∂Φ_j

∂pk

∂p_k

∂ql = −∂Φ_j

∂ql

, j, l = 1, . . . , n .

Replace now the second of these identities in the relation {Φj, Φ_m} = 0 , which holds true because the functions are assumed to be in involution. With a few calculations we get

{Φ^j, Φm} =X

l

 ∂Φ_j

∂ql

∂Φ_m

∂pl − ∂Φ_j

∂pl

∂Φ_m

∂ql



= −X

l,k

∂Φm

∂p_l

∂Φj

∂p_k

∂pk

∂q_l +X

l,k

∂Φj

∂p_l

∂Φm

∂p_k

∂pk

∂q_l

= −X

l

∂Φ_m

∂p_l X

k

∂Φ_j

∂p_k

 ∂p_k

∂q_l − ∂p_l

∂q_k



= 0

(note that in the second sum on the second line the indexes l and k can be exchanged).

By condition (3.13) this implies

∂p_k

∂q_l − ∂p_l

∂q_k = 0 , l, k = 1, . . . , n , so that the differential formP

jp_jdq_j is exact, as claimed. By integration we construct the generating function (3.14) which, in view of (3.13), satisfies the invertibility con- dition (2.29) of proposition 2.17. Therefore, the wanted canonical transformation is

implicitly defined by

α_j = ∂S

∂Φj

, p_j = ∂S

∂qj

, j = 1, . . . , n .

Q.E.D.

3.2 The theorem of Liouville

For a generic system of differential equations on a n–dimensional manifold a complete integration by quadrature can be performed when n − 1 independent first integrals are known, n being the dimension of the space. Thus, one expects that in the Hamiltonian case, the dimension of phase space being 2n, one needs to know 2n − 1 first integrals.

However, the canonical structure allows us to perform the complete integration if only n first integrals are known, provided they fulfill the further condition of being in involution.

Theorem 3.11: Assume that an autonomous canonical system with n degrees of freedom and with Hamiltonian H(q, p) possesses n independent first integral {Φ1(q, p), . . . , Φ_n(q, p)} forming a complete involution system. Then the system is integrable by quadratures. More precisely, one can construct the generating func- tion S(Φ, q) of a canonical transformation (q, p) = χ(α, Φ) such that the transformed Hamiltonian depends only on the new momenta Φ₁, . . . , Φ_n, and the solutions are expressed as

α_j(t) = α_j,0+ t∂H

∂Φ_j   

(Φ¹,0,...,Φn,0)

, j = 1, . . . , n , with αj,0 and Φj,0 determined by the initial data.

3.2.1 Proof of Liouville’s theorem

By proposition 3.8 we can construct by quadratures a canonical transformation (q, p) = χ(α, Φ) such that Φ₁, . . . , Φ_n are the new momenta. In view of preserva- tion of Poisson brackets, we can compute the Poisson bracket {H, Φj} with respect to the new variables α, Φ. Since Φ₁, . . . , Φ_n are first integrals, this gives

{H, Φj} = ∂H

∂α_j = 0 , j = 1, . . . , n .

This means that the transformed Hamiltonian depends only on the momenta, i.e., H = H(Φ). Therefore, the canonical equations are

˙α_j = ∂H

∂Φj

, Φ˙_j = 0 , j = 1, . . . , n , and are trivially integrable, as stated. This concludes the proof.

3.2.2 Integration procedure

Liouville’s theorem together with proposition 3.8 actually furnishes an explicit inte- gration algorithm. Here is the procedure.

(i) If necessary, exchange some pairs of canonical variables so that the condition det ∂(Φ₁, . . . , Φ_n)

∂(p₁, . . . , p_n)

 6= 0

is fulfilled. Then perform an inversion, finding p = p(Φ, q) . (ii) Construct the generating function

S(Φ, q) =

Z X

j

p_j(Φ, q)dq_j

and find the canonical coordinates α conjugated to Φ as αj = ∂S

∂Φ_j . This requires a quadrature.

(iii) By substitution, determine the transformed Hamiltonian H(Φ) . (iv) The solutions of the canonical equations are

(3.16) Φ_j(t) = Φ_j,0 , α_j(t) = ∂H

∂Φj

Φj=Φj,0

t + α_j,0 , j = 1, . . . , n , where Φ_j0 and α_j,0 are the initial values that can be computed from the initial data.

(v) By inversion of the canonical transformation find q = q(Φ, α) and p = p(Φ, α) . (vi) The solutions q(t), p(t) in the original variables are found by substitution of

Φ(t), α(t) given by (3.16).

Example 3.4:Systems with one degree of freedom. Let us consider the Hamiltonian H(x, p) = p²

2m + V (x) ,

describing the motion of a mass point on a straight line under the action of the potential V (x). The condition at point (i) reduces to ^∂H_∂p 6= 0, which is fulfilled for p 6= 0. Setting H(x, p) = E, we invert the relation above with respect to p, getting

(3.17) p = ±p2m[E − V (x)] .

The generating function is

S(E, x) =√ 2m

Z

pE − V (x)dx , and the canonical transformation in implicit form is written as

p = ±p2m[E − V (x)] , α =r m 2

Z dx

pE − V (x) .