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A Classical Mechanics Concepts
and {f, H } P oisson = −{H, f } P oisson . The Poisson bracket of any two phase
functions, f ({q i }, {p i }, t) and g({q i }, {p i }, t) is written
{f, g} P oisson =
3N
i=1
∂f
∂q i
∂g
∂p i
−
∂f
∂p i
∂g
∂q i
.
(A.12)
The Poisson bracket is invariant under canonical transformations. If we make a
canonical transformation from coordinates (p, q) to coordinates (P , Q) (that is,
p = p(P , Q), q = q(P , Q)), the Poisson bracket is given by Eq. (A.12) but with
p → P and q → Q and f = f (p(P , Q), q(P , Q)).
A.5 Phase Space Volume Conservation
One of the important properties of the Hamiltonian phase space is that volume
elements are conserved under the flow of points in phase space. A volume element
at some initial time t o can be written
dV
N
t o
= dp 1 (t o ) . . . dp 3N (t o )dq 1 (t o ) . . . dq 3N (t o ).
It is related to a volume element, dV t , at time t by the Jacobian, J 3N (t o , t), of the
transformation between phase space coordinates at time t o , {p i (t o )}, {q i (t o )}, and
coordinates at time t, {p i (t)}, {q i (t)}. Thus
dV
N
t = J 3N (t, t o )dV
N
t o
.
(A.13)
For systems obeying Hamilton’s equations (even if they have a time-dependent
Hamiltonian), the Jacobian is a constant of the motion,
dJ 3N (t, t o )
dt
= 0,
(A.14)
and therefore the size of volume elements in phase space does not change in time.
A.6 Action-Angle Coordinates
We can write Hamilton’s equations in terms of any convenient set of generalized
coordinates. We can transform between coordinate systems and leave the form of
Hamilton’s equations invariant via canonical transformations. There is, however,
one set of canonical coordinates, the action-angle variables, that play a distinctive
role both in terms of the analysis of chaotic behavior in classical nonlinear systems