§ 1. The algebra of observables in classical mechanics 3

For simplicity the exposition to follow is conducted using the example

of a system with one degree of freedom. The Hamiltonian equations

in this case have the form

(6) ˙ q =

∂H

∂p

, ˙ p = −

∂H

∂q

, H = H(q, p).

The Cauchy problem for the system (6) and the initial conditions

(7) q|t=0 = q0, p|t=0 = p0

has a unique solution

(8) q = q(q0, p0, t), p = p(q0, p0, t).

For brevity of notation a point (q, p) in phase space will sometimes

be denoted by µ, and the Hamiltonian equations will be written in

the form

(9) ˙ µ = v(µ),

where v(µ) is the vector field of these equations, which assigns to each

point µ of phase space the vector v with components

∂H

∂p

,

−∂H

∂q

.

The Hamiltonian equations generate a one-parameter commuta-

tive group of transformations

Gt : M → M

of the phase space into

itself,2

where Gtµ is the solution of the Hamil-

tonian equations with the initial condition Gtµ|t=0 = µ. We have the

equalities

(10) Gt+s = GtGs = GsGt, Gt

−1

= G−t.

In turn, the transformations Gt generate a family of transformations

Ut : A → A

of the algebra of observables into itself, where

(11) Utf(µ) = ft(µ) = f(Gtµ).

In coordinates, the function ft(q, p) is defined as follows:

(12) ft(q0, p0) = f(q(q0, p0, t), p(q0, p0, t)).

2We

assume that the Hamiltonian equations with initial conditions (7) have a

unique solution on the whole real axis. It is easy to construct examples in which

a global solution and, correspondingly, a group of transformations Gt do not exist.

These cases are not interesting, and we do not consider them.