A.4 The Poisson Bracket
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A.3 Hamilton’s Equations
In the Newtonian and Lagrangian formulations of mechanics, dynamical systems
are described in terms of a phase space composed of generalized velocities and
positions. The Hamiltonian formulation describes such systems in terms of a
phase space composed of generalized momenta, {p i }, and positions, {q i }. The
Hamiltonian phase space has very special properties. If the system has some
translational symmetry, then some of the momenta may be conserved quantities.
In addition, for systems obeying Hamilton’s equations of motion, the size of
volume elements in phase space is conserved. Thus the phase space behaves like
an incompressible fluid. A Legendre transformation from coordinates { ˙
q i }, {q i } to
coordinates {p i }, {q i } yields the following equations of motion for the Hamiltonian
phase space coordinates
˙
p i =
dp i
dt
= −
∂H
∂q i
,
(A.6)
˙
q i =
dq i
dt
=
∂H
∂p i
,
(A.7)
∂H
∂t
= −
∂L
∂t
.
(A.8)
Equations (A.6)–(A.8) are called Hamilton’s equations.
A.4 The Poisson Bracket
The equation of motion of any phase function (any function of phase space
variables) may be written in terms of Poisson brackets. Let us consider a phase
function, f ({q i }, {p i }, t). Its total time derivative is
df
dt
=
∂f
∂t
+
3N
i=1
∂f
∂q i
˙
q i +
∂f
∂p i
˙
p i
.
(A.9)
Using Hamilton’s equations, we can write this in the form
df
dt
=
∂f
∂t
+ {f, H } P oisson ,
(A.10)
where
{f, H } P oisson =
3N
i=1
∂f
∂q i
∂H
∂p i
−
∂f
∂p i
∂H
∂q i
(A.11)