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
∂pjp ̇j=−
∂h
∂qj
Specializing to the cased= 1, for any observable functionf, Hamilton’s
equations imply
df
dt=
∂f
∂qdq
dt+
∂f
∂pdp
dt=
∂f
∂q∂h
∂p−
∂f
∂p∂h
∂qWe 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
∂qAn 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
∂pp ̇={p,h}=−∂h
∂qFor a non-relativistic free particle,h=p2
2 mand these equations becomeq ̇=p
m, p ̇= 0