22 Classical mechanics
=
∑^3 N
α=1[
∂a
∂qαq ̇α+∂a
∂pαp ̇α]
=
∑^3 N
α=1[
∂a
∂qα∂H
∂pα−
∂a
∂pα∂H
∂qα]
≡ {a,H}. (1.6.19)The last line is known as thePoisson bracketbetweena(x) andH(x). The general
definition of a Poisson bracket between two functionsa(x) andb(x) is
{a,b}=∑^3 N
α=1[
∂a
∂qα∂b
∂pα−
∂a
∂pα∂b
∂qα]
. (1.6.20)
Note that the Poisson bracket is a statement about the dependence of functions on
the phase space vector and no longer refers to time. This is an important distinction,
as it will often be necessary for us to distinguish between quantitiesevaluated along
trajectories generated from the solution of Hamilton’s equations and quantities that
are evaluated at arbitrary (static) points in the phase space. From eqn. (1.6.20), it
is clear that ifa(x) is a conserved quantity, then da(xt)/dt= 0 along a trajectory,
and, therefore,{a(x),H(x)}= 0 in the phase space. Conversely, if the Poisson bracket
between any quantitya(x) and the Hamiltonian of a system vanishes, then the function
a(xt) is conserved along a trajectory generated by Hamilton’s equations.
As an example of the Poisson bracket formalism, suppose a system has no external
forces acting on it. In this case, the total force
∑N
i=1Fi= 0, since all internal forces
are balanced by Newton’s third law. The condition
∑N
i=1Fi= 0 implies that∑Ni=1Fi=−∑N
i=1∂H
∂ri= 0. (1.6.21)
Now, consider the total momentumP=
∑N
i=1pi. Its Poisson bracket with the Hamil-
tonian is
{P,H}=∑N
i=1{pi,H}=−∑N
i=1∂H
∂ri=
∑N
i=1Fi= 0. (1.6.22)Hence, the total momentumPis conserved. When a system has no external forces
acting on it, its dynamics will be the same no matter where in space thesystem lies.
That is, if all of the coordinates were translated by a constant vectoraaccording
tor′i=ri+a, then the Hamiltonian would remain invariant. This transformation
defines the so-calledtranslation group. In general, if the Hamiltonian is invariant with
respect to the transformations of a particular groupG, there will be an associated
conservation law. This fact, known asNoether’s theorem, is one of the cornerstones of
classical mechanics and also has important implications in quantum mechanics.