The evolution in time of physical systems is described by differential equations of the form
˙x = u(x) , (1.1)
where the vector field u :Γ3 x 7→ u(x) ∈ TxΓis defined on some phase space (i.e. the space of all the possible states of the system)Γ and takes on values in its tangent bundle TΓ:=S
x∈ΓTxΓ (i.e. the union of all the tangent spaces TxΓ). Notice that ifΓ= Rnthen TΓ=Γ= Rn as well.
Remark 1.1. From a topological point of view, the phase spaceΓof the system has to be at least a Banach space. This is due to the necessity to guarantee the existence and uniqueness of the solutionx(t) =Φt(x0) to the differential equation (1.1), for any initial condition x(0) = x0∈Γand anyt ∈ Ix0⊆ R; see Appendix A and reference [38].
Very often,Γ is a Hilbert space (something more than Banach). This happens ifΓis finite dimensional or, for example, in the theory of the classical linear PDEs, such as the wave and the heat equations. The most notable case of an infinite dimensional Hilbert phase space is perhaps that of quantum mechanics (see below).
Hamiltonian systems are those particular dynamical systems whose phase spaceΓ is en-dowed with a Poisson structure, according to the following definition.
Definition 1.1 (Poisson bracket). Let A(Γ) be the algebra (i.e. vector space with a bilinear product) of real smooth functions defined onΓ. A function { , } :A(Γ) ×A(Γ) →A(Γ) is called a Poisson bracket onΓif it satisfies the following properties:
1. {F, G} = −{G, F} ∀F,G ∈A(Γ) (skew-symmetry);
2. {αF + βG, H} = α{F, H} + β{G, H} ∀α,β ∈ R and ∀F,G, H ∈A(Γ) (left linearity);
3. {F, {G, H}} + {G,{H, F}} + {H,{F,G}} = 0 ∀F,G, H ∈A(Γ) (Jacobi identity);
4. {FG, H} = F{G, H} + {F, H}G ∀F,G, H ∈A(Γ) (left Leibniz rule).
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The pair (A(Γ), { , }) is a Poisson algebra, i.e. a Lie algebra (a vector space with a skew-symmetric, left-linear and Jacobi product) satisfying the Leibniz rule. Observe that 1. and 2.
in the above definition imply right-linearity, so that the Poisson bracket is actually bi-linear.
Observe also that 1. and 4. imply the right Leibniz rule.
Definition 1.2 (Hamiltonian system). Given a Poisson algebra (A(Γ), { , }), a Hamiltonian sys-tem on Γ is a dynamical system described by a differential equation of the form (1.1) whose vector field has the form
u(x) = XH(x) := {x, H} . whereH ∈A(Γ) is called the Hamiltonian of the system.
In the above definition the bracket {x, H} is meant by components with respect to the first entry: uk(x) = [XH(x)]k:= {xk, H}, where xk denotes the k-th component of x. Notice that k is not necessarily a discrete index. The following proposition holds:
Proposition 1.1. A skew-symmetric, bi-linear Leibniz bracket {, } onΓis such that
{F, G} = ∇F · J∇G :=X
The bracket at hand is Jacobi-like, i.e. it is a Poisson bracket, iff the operator function J(x) satisfies the relation:
CPROOF. Let us prove the proposition in the simple case of the algebraA(Γ) of the ana-lytic functions onΓ⊆ Rn. As a preliminary remark, we observe that from the Leibniz rule and from the linearity property it follows that {F, c} = 0 for any F ∈A(Γ) and any c ∈ R. Indeed,
1.1. POISSON STRUCTURES 9 From the last two points (1.2) easily follows. The condition (1.4) is checked by a direct compu-tation. One has
Now, exploiting the skew-symmetry of J and the consequent symmetry of a matrix of the form J(hessian)J, and suitably cycling over the functions F, G, H and over the indices i, j, k, one gets
Obviously, such an expression is identically zero for all F, G, H ∈A(Γ) iff (1.4) holds (show that).B
As a consequence of the above proposition, the Hamiltonian vector fields are of the form (put F = x and G = H in (1.3))
XH(x) := {x, H} = J(x)∇xH(x) , (1.5) i.e. are proportional to the gradient of the Hamiltonian function through the function operator J(x) := {x, x}. The latter operator, when (1.4) is satisfied, takes the name of Poisson tensor. A Poisson tensor J(x) is singular at x if there exists a vector field u(x) 6≡ 0 such that J(x)u(x) = 0, i.e. if ker J(x) is nontrivial. The functions C(x) such that ∇C(x) ∈ ker J(x) have a vanishing Poisson bracket with any other function F defined on Γ, since {F, C} = ∇F · J∇C ≡ 0 inde-pendently of F. Such special functions are called Casimir invariants associated to the given Poisson tensor J, and are constants of motion of any Hamiltonian system with vector field XH associated to J, i.e.:
Example 1.1. In the “standard case” x = (q, p), where q ∈ Rn is the vector of generalized co-ordinates and p ∈ Rn is the vector of the conjugate momenta. The Hamilton equations read q = ∂H/∂p, ˙p = −∂H/∂q, or, in compact form ˙x = J˙ 2n∇xH, where is the 2n × 2n standard symplectic matrix, independent of x. The Poisson bracket reads then {F, G} =Pn
j=1£(∂qjF)(∂pjG) − (∂qjG)(∂pjF)¤ = ∂F∂q ·∂G∂p−∂F∂p·∂G∂q. Everything extends to the case n → +∞, paying attention to matters of convergence.
We recall that the standard form of the Hamilton equations just mentioned is implied by that of the Euler-Lagrange equations when passing from the Lagrangian to the Hamiltonian formalism through a Legendre transformation, if this is possible.
Example 1.2. Let us consider a single harmonic oscillator, with Hamiltonian H =12¡ p2+ ω2q2¢, and introduce the complex variables z = (ωq + i p)/p
2ω and z∗= (ωq − i p)/p
2ω, where i is the imaginary unit. In terms of such complex coordinates, known as complex Birkhoff coordinates, the Hamiltonian reads H = ω|z|e 2, the Hamilton equations become ˙z = −iωz = −i∂ eH/∂z∗ and its complex conjugate ˙z∗= iωz = i∂ eH/∂z. The new Poisson tensor is the second Pauli matrix, σ2=
Example 1.3. The Euler equations, describing the evolution of the angular momentum L of a rigid body in a co-moving frame and in absence of external torque (moment of external forces), read ˙L = L ∧ I−1L, where I is the inertia tensor of the body (a 3 × 3 symmetric, positive definite matrix) and ∧ denotes the standard vector product in R3. This is a Hamiltonian system with Hamiltonian functionH(L) =12L · I−1L and Poisson tensor
J(L) :=
In this way, the Euler equations have the standard form ˙L = J(L)∇LH(L), and the Poisson bracket of two functions F(L) and G(L) is {F, G} = L · (∇G ∧ ∇F). In order to check whether the relation (1.4) holds, observe that one can write Ji j(L) = −P3
k=1εi jkLk, where εi jk is the Levi-Civita symbol with three indices i, j, k = 1,2,3, so defined: εi jk= +1 if (i, j, k) is an even permutation of (1, 2, 3);εi jk= −1 if (i, j, k) is an odd permutation of (1, 2, 3) and εi jk= 0 if any two indices are equal (recall that a permutation is even or odd when it is composed by an even or odd number of pair exchanges, respectively). The following relation is also useful: εi jkεilm= δjlδkm− δklδjm, whereδi j is the Kronecker delta, whose value is 1 if i = j and zero otherwise.
We finally notice that the Casimir invariants of J(L) are all the functions C(L) := f¡|L|2¢, since
∇C = f0¡|L|2¢ 2L, so that J(L)∇C = L ∧ (2f0L) = 0.
Example 1.4. The wave equation utt = uxx, where the unknown function u(t, x) is defined on R × [0,`] with periodic boundary conditions, i.e. u is actually defined on R × R/(`Z). The wave equation is Hamiltonian, with Hamiltonian function given by the enrgy integral, namely H(u, p) = 12R`
0 ¡ p2+ u2x¢ dx and Poisson tensor J2, the standard 2 × 2 symplectic matrix. In this way, the Hamilton equations read ut= δH/δp = p, pt = −δH/δu = uxx. Indeed the gradient
∇H = (δH/δu, δH/δp) is meant in the L2 sense, i.e. ∇H is the object that multiplies the incre-ment in the differential ofH:
dH(u, p; h, k) := d
In this case the phase space Γ of the system is the set of pairs of space-periodic functions (u, p)(t, x), with some specified regularity. Observing that u enters the Hamiltonian H only
1.2. CANONICAL TRANSFORMATIONS 11