there is oscillating motion by laser. This force by the nearest ion appears randomly,
and the velocity space of the electron shows the spread with time like diffusion in
velocity (energy) space as shown in Chap. 2.
8.4.2 Simple Example [2] (Periodic Force)
It is said that the Poincare pointed out in 1892 that three celestial body motions
cannot be integrated and the motions are not predictable. This is the origin of the
study of chaos in Hamiltonian system. It is good example to consider the orbit of an
asteroid rotating around the sun as shown in Fig. 8.12a [5]. Such a two celestial body
problem is solved analytically. According to the solar system formation theory, it is
expected there are many asteroids in the solar system. It is not, however, observed in
the orbit between Mars and Jupiter as expected from two body motion. It is a good
example to know how an integrable asteroid orbit around the sun becomes unstable
by the periodic perturbation due to the gravitational force by Jupiter. Here, “unstable” means the total energy of the asteroid becomes positive due to the periodic force
by Jupiter to escape from the solar system. In Fig. 8.12b, observation data of the
number of asteroids are plotted as a function of the oscillation frequency of asteroid
normalized by that of Jupiter. The orbit frequency of asteroids is a simple function of
the mean radius of the orbit, and the frequency is a function of the orbit radius. The
left is near Mars and the right is near Jupiter in Fig. 8.12b.
Sun
Asteroid
Jupiter
(a)
(b)
f
1
2
3
4
ω/ω 3
ω 3
w
Fig. 8.12 (a) Schematics of an asteroid orbit bounded by the gravity of the sun. The Jupiter gravity
affects the Kepler motion of the asteroid as a periodic perturbation force. (b) The observation data of
the number of asteroids between Mars and Jupiter. It is inferred that due to the Jupiter perturbation
(frequency ω J ) the asteroids rotating with the frequency ω become unstable, and the kinetic energy
becomes larger than the binding energy by the sun to escape from the solar system
8.4 Chaotic Motion due to External Force
307
and the velocity space of the electron shows the spread with time like diffusion in
velocity (energy) space as shown in Chap. 2.
8.4.2 Simple Example [2] (Periodic Force)
It is said that the Poincare pointed out in 1892 that three celestial body motions
cannot be integrated and the motions are not predictable. This is the origin of the
study of chaos in Hamiltonian system. It is good example to consider the orbit of an
asteroid rotating around the sun as shown in Fig. 8.12a [5]. Such a two celestial body
problem is solved analytically. According to the solar system formation theory, it is
expected there are many asteroids in the solar system. It is not, however, observed in
the orbit between Mars and Jupiter as expected from two body motion. It is a good
example to know how an integrable asteroid orbit around the sun becomes unstable
by the periodic perturbation due to the gravitational force by Jupiter. Here, “unstable” means the total energy of the asteroid becomes positive due to the periodic force
by Jupiter to escape from the solar system. In Fig. 8.12b, observation data of the
number of asteroids are plotted as a function of the oscillation frequency of asteroid
normalized by that of Jupiter. The orbit frequency of asteroids is a simple function of
the mean radius of the orbit, and the frequency is a function of the orbit radius. The
left is near Mars and the right is near Jupiter in Fig. 8.12b.
Sun
Asteroid
Jupiter
(a)
(b)
f
1
2
3
4
ω/ω 3
ω 3
w
Fig. 8.12 (a) Schematics of an asteroid orbit bounded by the gravity of the sun. The Jupiter gravity
affects the Kepler motion of the asteroid as a periodic perturbation force. (b) The observation data of
the number of asteroids between Mars and Jupiter. It is inferred that due to the Jupiter perturbation
(frequency ω J ) the asteroids rotating with the frequency ω become unstable, and the kinetic energy
becomes larger than the binding energy by the sun to escape from the solar system
8.4 Chaotic Motion due to External Force
307