§ 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) ˙ = ∂H ∂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) ˙ = 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, G−1 t = 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)). 2 We 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.
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