Linear Beam Optics
43
later
eventually
FIGURE 2.8: Poincar´ e’s recurrence theorem. After sufficient time, the
system returns close to its original state.
tracking pictures. Even for our daily life, there are important consequences.
If the universe is Hamiltonian and it does not expand indefinitely, then up to
minute details, history will keep repeating itself. So we will all be born again,
and we will all make the same mistakes all over, but since now we cannot
remember anything about our past life, also next time we will not remember
our current life.
Now let us sketch the proof of the recurrence theorem. Let an ε be given,
and consider an ε-ball with volume V ε in phase space. Consider its motion by
regular time steps Δt. Since the total available phase space volume is finite,
say V p , after at most V p /V ε time steps, the image of the ball must reach a
part of phase space it has touched before, i.e., it must overlap a previous
image of the ball. Let us assume this happens after N steps and the previous
image is that after n steps, with n < N. But if the images after n steps I n
and after N steps I N overlap, so must the images after (n − 1) and (N − 1)
steps, respectively. And continuing backwards, so must the images after 0 and
(N − n) steps; hence, after (N − n) steps, we touch the original ε-ball again.
2.3 Special Optical Systems
In this section we want to apply the matrix techniques to the study of
certain special categories of systems. In particular, we associate certain fundamental properties of systems with properties of the matrix. We begin with
the imaging systems.
43
later
eventually
FIGURE 2.8: Poincar´ e’s recurrence theorem. After sufficient time, the
system returns close to its original state.
tracking pictures. Even for our daily life, there are important consequences.
If the universe is Hamiltonian and it does not expand indefinitely, then up to
minute details, history will keep repeating itself. So we will all be born again,
and we will all make the same mistakes all over, but since now we cannot
remember anything about our past life, also next time we will not remember
our current life.
Now let us sketch the proof of the recurrence theorem. Let an ε be given,
and consider an ε-ball with volume V ε in phase space. Consider its motion by
regular time steps Δt. Since the total available phase space volume is finite,
say V p , after at most V p /V ε time steps, the image of the ball must reach a
part of phase space it has touched before, i.e., it must overlap a previous
image of the ball. Let us assume this happens after N steps and the previous
image is that after n steps, with n < N. But if the images after n steps I n
and after N steps I N overlap, so must the images after (n − 1) and (N − 1)
steps, respectively. And continuing backwards, so must the images after 0 and
(N − n) steps; hence, after (N − n) steps, we touch the original ε-ball again.
2.3 Special Optical Systems
In this section we want to apply the matrix techniques to the study of
certain special categories of systems. In particular, we associate certain fundamental properties of systems with properties of the matrix. We begin with
the imaging systems.