§ 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.