30
2 Fundamental Concepts
We can again make a canonical transformation, J 1 = J 1 −
2
3 J 2 , J 2 = J 2 , θ 1 =
1 , θ 2 = 2 +
2
3 1 (note that I = 3J 1 ). The Hamiltonian then takes the form
H = J 1 − J
2
1 +
J 2
3
−
5J 1 J 2
3
+
23
9
J
2
2 +
β
3
J
3
2
2 (3J 1 − 2J 2 ) cos(3 2 ) = E
(2.55)
and the coordinate J 1 is a constant of the motion since H is independent of 1 . The
equations of motion for J 2 and 2 are
dJ 2
dt
= βJ
3
2
2 (3J 1 − 2J 2 ) sin(3 2 )
(2.56)
and
dd 2
dt
=
1
3
−
5J 1
3
+
46J 2
9
+ βJ
1
2
2
3
2
J 1 −
5
3
J 2
cos(3 2 ).
(2.57)
It is easy to see that the fixed points occur for 2 =
nπ
3 and J 2 = J o where J o
satisfies the equation
1
3
−
5I
9
+
46J o
9
+ βJ
1
2
o
I
2
−
5
3
J o
cos(nπ ) = 0.
(2.58)
If we again linearize the equations of motion about these fixed points and determine
the form of the flow in their neighborhood as we did below Eq. (2.45), we find that
for even n (n = 0, 2, 4) the fixed points are hyperbolic while for odd n (n = 1, 3, 5)
the fixed points are elliptic. These fixed points are clearly seen in the plot of the
phase space trajectories for the (2,3) resonance system given in Fig. 2.7. In Fig. 2.7
all curves have energy E = 0.18 and coupling constant β = 0.1. The separatrix of
the (2,3) resonance zone is clearly seen, as are the three hyperbolic and elliptic fixed
points.
2.5.2 Two-Resonance Hamiltonian
The two single-resonance systems described above are integrable. Any systems
containing two or more resonances are nonintegrable because a second isolating
integral of the motion cannot be found. Therefore systems with two or more
resonances can undergo a transition to chaos as parameters of the system are varied.
Walker and Ford showed this for the Hamiltonian with two primary resonances,
H = H o (J 1 , J 2 ) + αJ 1 J 2 cos(2θ 1 − 2θ 2 )
+βJ 1 J
3
2
2 cos(2θ 1 − 3θ 2 ) = E.
(2.59)
2 Fundamental Concepts
We can again make a canonical transformation, J 1 = J 1 −
2
3 J 2 , J 2 = J 2 , θ 1 =
1 , θ 2 = 2 +
2
3 1 (note that I = 3J 1 ). The Hamiltonian then takes the form
H = J 1 − J
2
1 +
J 2
3
−
5J 1 J 2
3
+
23
9
J
2
2 +
β
3
J
3
2
2 (3J 1 − 2J 2 ) cos(3 2 ) = E
(2.55)
and the coordinate J 1 is a constant of the motion since H is independent of 1 . The
equations of motion for J 2 and 2 are
dJ 2
dt
= βJ
3
2
2 (3J 1 − 2J 2 ) sin(3 2 )
(2.56)
and
dd 2
dt
=
1
3
−
5J 1
3
+
46J 2
9
+ βJ
1
2
2
3
2
J 1 −
5
3
J 2
cos(3 2 ).
(2.57)
It is easy to see that the fixed points occur for 2 =
nπ
3 and J 2 = J o where J o
satisfies the equation
1
3
−
5I
9
+
46J o
9
+ βJ
1
2
o
I
2
−
5
3
J o
cos(nπ ) = 0.
(2.58)
If we again linearize the equations of motion about these fixed points and determine
the form of the flow in their neighborhood as we did below Eq. (2.45), we find that
for even n (n = 0, 2, 4) the fixed points are hyperbolic while for odd n (n = 1, 3, 5)
the fixed points are elliptic. These fixed points are clearly seen in the plot of the
phase space trajectories for the (2,3) resonance system given in Fig. 2.7. In Fig. 2.7
all curves have energy E = 0.18 and coupling constant β = 0.1. The separatrix of
the (2,3) resonance zone is clearly seen, as are the three hyperbolic and elliptic fixed
points.
2.5.2 Two-Resonance Hamiltonian
The two single-resonance systems described above are integrable. Any systems
containing two or more resonances are nonintegrable because a second isolating
integral of the motion cannot be found. Therefore systems with two or more
resonances can undergo a transition to chaos as parameters of the system are varied.
Walker and Ford showed this for the Hamiltonian with two primary resonances,
H = H o (J 1 , J 2 ) + αJ 1 J 2 cos(2θ 1 − 2θ 2 )
+βJ 1 J
3
2
2 cos(2θ 1 − 3θ 2 ) = E.
(2.59)