§ 1. The algebra of observables in classical mechanics 5 The properties 1), 2), and 4) follow directly from the definition of the Poisson brackets. The property 4) shows that the “Poisson bracket” operation is a derivation of the algebra of observables. In- deed, the Poisson bracket can be rewritten in the form {f, g} = Xf g, where Xf = ∂f ∂p ∂q ∂f ∂q ∂p is a first-order linear differential operator, and the property 4) has the form Xf gh = (Xf g)h + gXf h. The property 3) can be verified by differentiation, but it can be proved by the following argument. Each term of the double Poisson bracket contains as a factor the second derivative of one of the functions with respect to one of the variables that is, the left-hand side of 3) is a linear homogeneous function of the second derivatives. On the other hand, the second derivatives of h can appear only in the sum {f,{g, h}} + {g,{h, f}} = (Xf Xg XgXf )h, but a commutator of first-order linear differential operators is a first-order differential operator, and hence the second derivatives of h do not appear in the left-hand side of 3). By symmetry, the left-hand side of 3) does not contain second derivatives at all that is, it is equal to zero. The Poisson bracket {f, g} provides the algebra of observables with the structure of a real Lie algebra.3 Thus, the set of observables has the following algebraic structure. The set A is: 1) a real linear space 2) a commutative algebra with the operation fg 3) a Lie algebra with the operation {f, g}. The last two operations are connected by the relation {f, gh} = {f, g}h + g{f, h}. The algebra A of observables contains a distinguished element, namely, the Hamiltonian function H, whose role is to describe the 3 We recall that a linear space with a binary operation satisfying the conditions 1)–3) is called a Lie algebra.
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