24 Classical mechanics
structure. Consider a solution xtto eqn. (1.6.26) starting from an initial condition x 0.
Because the solution of Hamilton’s equations is unique for each initial condition, xt
will be a unique function of x 0 , that is, xt= xt(x 0 ). This dependence can be viewed
as defining a variable transformation on the phase space from an initial set of phase
space coordinates x 0 to a new set xt. The Jacobian matrix J of this transformation,
whose elements are given by
Jkl=∂xkt
∂xl 0, (1.6.28)
satisfies the following condition:
M = JTMJ, (1.6.29)
where JTis the transpose of J. Eqn. (1.6.29) is known as thesymplectic property.
We will have more to say about the symplectic property in Chapter 3.At this stage,
however, let us illustrate the symplectic property in a simple example.Consider, once
again, the harmonic oscillatorH=p^2 / 2 m+kx^2 /2 with equations of motion
x ̇=∂H
∂p=
p
mp ̇=−∂H
∂x=−kx. (1.6.30)The general solution to these for an initial condition (x(0),p(0)) is
x(t) =x(0) cosωt+p(0)
mωsinωt
p(t) =p(0) cosωt−mωx(0) sinωt, (1.6.31)whereω=
√
k/mis the frequency of the oscillator. The Jacobian matrix is, therefore,J =
∂x(t)
∂x(0)∂x(t)
∂p(0)
∂p(t)
∂x(0)∂p(t)
∂p(0)
=
cosωt mω^1 sinωt−mωsinωt cosωt
. (1.6.32)
For this two-dimensional phase space, the matrix M is given simply by
M =
(
0 1
−1 0
)
. (1.6.33)
Thus, performing the matrix multiplication JTMJ, we find
JTMJ =
cosωt −mωsinωt1
mωsinωt cosωt
0 1
−1 0
cosωt mω^1 sinωt−mωsinωt cosωt
=
cosωt −mωsinωt1
mωsinωt cosωt
−mωsinωt cosωt−cosωt −mω^1 sinωt