§ 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
3We
recall that a linear space with a binary operation satisfying the conditions
1)–3) is called a Lie algebra.
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