is known as “phase space”. Points in phase space can be thought of as uniquely
parametrizing possible initial conditions for classical trajectories, so another in-
terpretation of phase space is that it is the space that uniquely parametrizes
solutions of the equations of motion of a given classical mechanical system. The
basic axioms of Hamiltonian mechanics can be stated in a way that parallels
the ones for quantum mechanics.
Axiom(States).The state of a classical mechanical system is given by a point
in the phase spaceM=R^2 d, with coordinatesqj,pj, forj= 1,...,d.
Axiom(Observables).The observables of a classical mechanical system are the
functions on phase space.
Axiom(Dynamics).There is a distinguished observable, the Hamiltonian func-
tionh, and states evolve according to Hamilton’s equations
q ̇j=
∂h
∂pj
p ̇j=−
∂h
∂qj
Specializing to the cased= 1, for any observable functionf, Hamilton’s
equations imply
df
dt
=
∂f
∂q
dq
dt
+
∂f
∂p
dp
dt
=
∂f
∂q
∂h
∂p
−
∂f
∂p
∂h
∂q
We can define:
Definition(Poisson bracket).There is a bilinear operation on functions on the
phase spaceM=R^2 (with coordinates(q,p)) called the Poisson bracket, given
by
(f 1 ,f 2 )→{f 1 ,f 2 }=
∂f 1
∂q
∂f 2
∂p
−
∂f 1
∂p
∂f 2
∂q
An observablefevolves in time according to
df
dt
={f,h}
This relation is equivalent to Hamilton’s equations since it implies them by
takingf=qandf=p
q ̇={q,h}=
∂h
∂p
p ̇={p,h}=−
∂h
∂q
For a non-relativistic free particle,h=p
2
2 mand these equations become
q ̇=
p
m
, p ̇= 0