Hamiltonian formulation 23Another fundamental property of Hamilton’s equations is known asthe condi-
tion ofphase space incompressibility. To understand this condition, consider writing
Hamilton’s equations directly in terms of the phase space vector as
̇x =η(x), (1.6.23)whereη(x) is a vector function of the phase space vector x. Since
x = (q 1 ,...,q 3 N,p 1 ,...,p 3 N),it follows that
η(x) =(
∂H
∂p 1,...,
∂H
∂p 3 N,−
∂H
∂q 1,...,−
∂H
∂q 3 N)
. (1.6.24)
Eqn. (1.6.23) illustrates the fact that the general phase space “velocity” ̇x is a function
of x, suggesting that motion in phase space described by eqn. (1.6.23) can be regarded
as a kind of “flow field” as in hydrodynamics, where the flow pattern of a fluid is
described by a velocity fieldv(r). Thus, at each point in phase space, there will be a
velocity vector ̇x(x) equal toη(x). In hydrodynamics, the condition forincompressible
flow is that there be no sources or sinks in the flow, which means thatthe flow field is
divergence free:∇·v(r) = 0. In phase space flow, the analogous condition is∇x· ̇x(x) =
0, where∇x=∂/∂x is the phase space gradient operator. Hamilton’s equations of
motion guarantee that the incompressibility condition in phase spaceis satisfied. To
see this, consider the compressibility in generalized coordinates
∇x· ̇x =∑^3 N
α=1[
∂p ̇α
∂pα+
∂q ̇α
∂qα]
=
∑^3 N
α=1[
−
∂
∂pα∂H
∂qα+
∂
∂qα∂H
∂pα]
=
∑^3 N
α=1[
−
∂^2 H
∂pα∂qα+
∂^2 H
∂qα∂pα]
= 0, (1.6.25)
where the second line follows from Hamilton’s equations of motion.
One final important property of Hamilton’s equations that merits comment is the
so-calledsymplectic structureof the equations of motion. Given the form of the vector
function,η(x), introduced above, it follows that Hamilton’s equations can be recast as
̇x = M∂H
∂x, (1.6.26)
where M is a matrix expressible in block form as
M =(
0 I
−I 0
)
, (1.6.27)
where 0 andIare the 3N× 3 N zero and identity matrices, respectively. Dynami-
cal systems expressible in the form of eqn. (1.6.26) are said to possess a symplectic