44
2 Fundamental Concepts
Fig. 2.16 The phase space of
a hard-sphere gas is a K-flow.
Neighboring trajectories
diverge rapidly due to
collisions with the hard
convex surfaces
Fig. 2.17 The stadium billiard is a two-dimensional billiard with spherical ends of radius r
separated by parallel sides of length 2a. For a = 0, the motion of a billiard is integrable, but
for a > 0 it is a K-flow (Bunimovich 1974)
Hamiltonian of this system is not smooth and differentiable. The convex surface of
the barrier causes neighboring trajectories to exponentially diverge from one another
in phase space.
Let us now consider the dynamics of the Sinai billiard (Berry 1978). Since the
box has periodic boundary conditions, we may also view this system as that of
a particle moving through a lattice of circular barriers. Assume that the average
distance traveled between collisions with the barriers is D and the radius of the
pillars is R. If two neighboring trajectories (we assume they have the same velocity)
strike a barrier at points a distance 0 apart, the angular distance will be 0 =
0 /R (see Fig. 2.16). However, when they strike the next barrier a distance D
away, their points of collision will be separated a distance 1 ≈ 0 D, and the
angular separation of the collision points will be 1 ≈
D
R θ 0 . If we continue
this process for n collisions, the approximate angular spread of points of collision
will be n ≈
D
R
n θ 0 . The number of collisions, n, needed for a divergence
of one radian is n = ln((θ 0 )/ ln
R
D
. It is interesting to consider an example. Let
0 = 0.0001 radians and R/D = 0.1. Then n = 4 and it requires only four
collisions to achieve a divergence of one radian.
Another system that has been proven to be a K-flow is that of a billiard moving
in a planar concave region called a stadium. The stadium billiard consists of two
half circles of radius r connected by equal parallel line segments of length 2a (see
Fig. 2.17). When a = 0 and the system is circular, the motion of the billiard is
integrable. However, for a > 0, it becomes a K-flow, as was proved by Bunimovich
2 Fundamental Concepts
Fig. 2.16 The phase space of
a hard-sphere gas is a K-flow.
Neighboring trajectories
diverge rapidly due to
collisions with the hard
convex surfaces
Fig. 2.17 The stadium billiard is a two-dimensional billiard with spherical ends of radius r
separated by parallel sides of length 2a. For a = 0, the motion of a billiard is integrable, but
for a > 0 it is a K-flow (Bunimovich 1974)
Hamiltonian of this system is not smooth and differentiable. The convex surface of
the barrier causes neighboring trajectories to exponentially diverge from one another
in phase space.
Let us now consider the dynamics of the Sinai billiard (Berry 1978). Since the
box has periodic boundary conditions, we may also view this system as that of
a particle moving through a lattice of circular barriers. Assume that the average
distance traveled between collisions with the barriers is D and the radius of the
pillars is R. If two neighboring trajectories (we assume they have the same velocity)
strike a barrier at points a distance 0 apart, the angular distance will be 0 =
0 /R (see Fig. 2.16). However, when they strike the next barrier a distance D
away, their points of collision will be separated a distance 1 ≈ 0 D, and the
angular separation of the collision points will be 1 ≈
D
R θ 0 . If we continue
this process for n collisions, the approximate angular spread of points of collision
will be n ≈
D
R
n θ 0 . The number of collisions, n, needed for a divergence
of one radian is n = ln((θ 0 )/ ln
R
D
. It is interesting to consider an example. Let
0 = 0.0001 radians and R/D = 0.1. Then n = 4 and it requires only four
collisions to achieve a divergence of one radian.
Another system that has been proven to be a K-flow is that of a billiard moving
in a planar concave region called a stadium. The stadium billiard consists of two
half circles of radius r connected by equal parallel line segments of length 2a (see
Fig. 2.17). When a = 0 and the system is circular, the motion of the billiard is
integrable. However, for a > 0, it becomes a K-flow, as was proved by Bunimovich