variables (v, p) the wave equation reads vt= px, pt= vx, which are the Hamilton equations is the first Pauli matrix. A third Poisson structure is obtained introducing the new variables w±:= (v ± p)/p
. Such a Poisson structure of the wave equation is also useful to solve it, since the two first order equations are decoupled in this case, and one hasw±(t, x) = w±(0, x ± t), i.e. a left (+ sign) and right (− sign) translation of the initial profiles, respectively. As a final remark, notice that in all the three mentioned Poisson structures the Poisson tensor is constant (i.e. does not depend on the “point” in the phase space); as a consequence relation (1.4) is trivially satisfied.
Example 1.5. The Korteweg-de Vries (KdV) equation vt= −2γvxxx+ 6vvx is a one-dimensional model for the propagation of shallow water surface waves, and, as such, is also the most basic model for the dynamics of Tsunamis. The unknown functionv(t, x) is defined onR × [0,`] with periodic boundary conditions. The KdV equation is a Hamiltonian system with Hamiltonian function H(v) =R`
0
¡γv2x+ v3¢ dx and Poisson tensor J := ∂/∂x, the derivation operator, so that vt= J∇H = ∂/∂xδH/δv. Notice that J is skew-symmetric and does not depend on v, so that the relation (1.4) is trivially satisfied. Finally, notice that ∂/∂xδC/δv = 0 iff δC/δv is independent of x, which implies that the Casimir invariants of the system are C(v) =R`
0 vdx, as well as any function of it.
1.2 Canonical transformations
Given the system ˙x = J(x)∇xH(x), under whatever change of coordinates x 7→ y = f (x) with inverse y 7→ x = g(y), it transforms to ˙y = ˜J( y)∇yH( y), where ˜˜ H( y) := H(g(y)) and
the superscript T denoting transposition. The above formula turns out to be useful also when expressed in terms of x, which yields
J(x) := ˜ˆ J( f (x)) =
It can be shown that ˜J( y) is a Poisson tensor iff J(x) is a Poisson tensor, so that, starting from a Hamiltonian system and performing any change of variables, one still gets a Hamiltonian system. Among all the possible changes of variables, a privileged role is played by those leaving the Poisson tensor invariant in form, namely ˜J( y) = J(y) in (1.7), or ˆJ(x) = J(x) in (1.8). Such particular transformations are called canonical. Noncanonical transformations appear in the examples 1.2 and 1.4 above. Canonical transformation are particularly useful in the standard
Hamiltonian case, where the reference Poisson tensor is the standard symplectic matrix J2n which is equivalent to the relations
" ∂Q
for all i, j = 1,..., n. Such relations are the necessary and sufficient conditions for a change of variables to be canonical in standard Hamiltonian mechanics.
Example 1.6. Consider again the harmonic oscillator of example 1.2. The transformation (q, p) 7→ (φ, I) defined by q =p
2I/ωsinφ, p =p
2ωI cosφ, is canonical, since {φ, I}q,p= 1. The new Hamiltonian readsH(e φ, I) = ωI, and the corresponding equations read ˙φ = ω, ˙I = 0.
Sometimes, the requirement of canonicity in the sense just stated turns out to be too re-strictive. For example, the simple re-scaling like
(q, p, H, t) 7→ (Q, P, K, T) = (αq,βp,γH,δt) , (1.12) depending on four real parameters α,β,γ,δ, preserves the form of the Hamilton equations, namely dQ/dT = ∂K/∂P, dP/dT = −∂K/∂Q, under the unique condition αβ = γδ. On the other hand, in order to satisfy the relations (1.11) one needs the further condition αβ = 1, which appears a bit superfluous. In order to extend the concept of canonical transformations (in the standard case) one starts from the following Hamilton variational principle.
Proposition 1.2. The solutions of the Hamilton equations ˙q = ∂H/∂p, ˙p = −∂H/∂q are the critical points (i.e. curves) of the action functional
A(q, p) :=
The boundary term above vanishes, and dA = 0 for all the directions (h, k) iff the Hamilton equations hold.B
1.2. CANONICAL TRANSFORMATIONS 13 Consider now a transformation (q, p, H, t) 7→ (Q, P, K, T), such that T depends only on t and satisfying
p · dq − Hdt = c (P · dQ − KdT) + dF(q,Q, t) , (1.14) where c is a constant; p · dq − Hdt is the so-called Poincaré-Cartan 1-form (differential form of degree one). Integrating the relation (1.14) over [t1, t2] yields
Z t2
which we rewrite in short, with the notation indicated above, as Aold= cAnew+∆F. Suppos-ing now that the transformation at hand is such that to fixed values of q(t1) and q(t2) there correspond fixed values of Q(t1) and Q(t2), one has
Proposition 1.3. The transformation (q, p, H, t) 7→ (Q, P, K, T), satisfying (1.14), maps Hamil-ton equations into HamilHamil-ton equations.
CPROOF. dAold= cdAnew+d∆F, and the last term vanishes, so that dAold= 0 iff dAnew= 0.
According to (1.13), the condition dAnew/old= 0 is equivalent to the Hamilton equations. B Very often the canonical transformations are defined as those satisfying (1.14). The con-stant c and the function F(q, Q, t) appearing on the right hand side of (1.14) are called the valence and the generating function of the transformation.
Example 1.7. The rescaling (1.12) is canonical in the sense just stated with c = 1/(αβ) and F equal to any constant in (1.14), with the conditionαβ = γδ.
Let us now restrict our attention to the case of transformations that are canonical in the sense of (1.15), having unitary valence, i.e. c = 1, and not involving any re-parametrization of time, i.e. T(t) ≡ t. Notice that if the generating function F is not constant, then (1.14) implies
∂F
∂q = p ; ∂F
∂Q= P ; ∂F
∂t = K − H .
Starting from (1.14), one can introduce another generating function S(q, P, t) := F(q,Q, t)+Q·P, satisfying depends (at most) on time only, being independent of Q and P. This amounts to look for a canonical transformation such that the new Hamiltonian variables do not evolve in time.
Notice that one can setϕ(t) ≡ 0 without any loss of generality. Indeed, if S satisfies (1.16) with K = ϕ(t), then ˜S := S −R
ϕ(t)dt satisfies (1.16) with K ≡ 0. With this in mind, the first and the third of relations (1.16) yield the Hamilton-Jacobi equation
∂S
a first order PDE in the unknown function S(q, t). Notice that, among the possible solutions of equation (1.17), we are interested to the so-called complete integrals, namely those solutions depending on n parameters P1, . . . , Pnand such that
A complete integral of the Hamilton-Jacobi equation generates a canonical transformation (q, p, H, t) 7→ (Q, P,0, t), since condition (1.18), together with the first two equations of (1.16), allows to get (q, p) in terms of (Q, P) and t and/or viceversa.
Very often, especially in problems of perturbation theory, H is independent of time, and one looks for time-independent canonical transformations such that the new Hamiltonian K de-pends on the momenta P only, since in the latter case the Hamilton equations are immediately solved: Q(t) = Q(0) + t(∂K/∂P), at constant P (such canonical transformations rectify the flow of the given Hamiltonian system). In this case, the following time-independent version of the Hamilton-Jacobi equation holds:
Here again, a complete integral of the above equation is required. Notice that the new Hamil-tonian K is an unknown of the problem.
Another very convenient way of performing canonical transformations is to do it through Hamiltonian flows. More precisely, let us consider a Hamiltonian G(Q, P) and its associ-ated Hamilton equations ˙Q = ∂G/∂P, ˙P = −∂G/∂Q. Let ΦGs denote the flow of G, so that (Q, P) =ΦsG(q, p) is the solution of the Hamilton equations at time s, corresponding to the initial condition (q, p) at s = 0. We also denote by
LG:= {·,G} = (J∇G) · ∇ = XG· ∇ (1.20) the Lie derivative along the Hamiltonian vector field XG; notice that LGF = {F,G}.
Proposition 1.4. The change of variablesΦGs : (q, p) 7→ (Q, P) is canonical.
C PROOF. We want to show that (1.9)-(1.10) are satisfied. Due to a repetitive use of the Leibniz rule, one has
1.3. INTEGRABILITY: LIOUVILLE THEOREM 15