34
2 Fundamental Concepts
2.6.1 The KAM theorem (for N = 2)
Consider a system described by the Hamiltonian
H (J 1 , J 2 , θ 1 , θ 2 ) = H o (J 1 , J 2 ) +
∞
n 1 =−∞
∞
n 2 =−∞
V n 1 ,n 2 (J 1 , J 2 )e
i(n 1 θ 1 +n 2 θ 2 ) ,
(2.66)
where is a small parameter and H 0 has nonzero Hessian. The prime on the
summations indicates that we exclude the term n 1 = n 2 = 0 since it can be included
in H 0 . We shall assume that H is an analytic function of all variables and is a
periodic function of angles θ 1 and θ 2 . On a torus (J 1 = J o
1 , J 2 = J o
2 ) such that
the frequencies ω i = (
∂H o
∂J j
) o = ω i (J o
1 , J o
2 ) satisfy the conditions
|n 1 ω 1 + n 2 ω 2 | ≥
K
||n|| α ,
where ||n|| = |n 1 | + |n 2 | > 0, α ≥ 2, and K is a constant, a perturbation theory
will converge.
The proof of the KAM theorem proceeds as follows (see Kolmogorov 1954 and
Barrar 1970 for details). Let us move the origin of the coordinates to (J o
1 , J o
2 ) via a
canonical transformation, J i − J o
i = p i and θ i = φ i . The Hamiltonian can then be
written in the form
H = C
(0)
+
2
i=1
ω i p i + A
(0) (φ 1 , φ 2 ) +
2
i=1
B
(0)
i (φ 1 , φ 2 )p i
+
2
i=1
2
j =1
C
(0)
i,j (φ 1 , φ 2 )p i p j + D
(0) (p 1 , p 2 , φ 1 , φ 2 ),
(2.67)
where C (0) is a constant and D (0) (p 1 , p 2 , φ 1 , φ 2 ) is a function whose lowestorder dependence on p i is p 3
i . Let us now introduce a generating function that
takes us from coordinates (p 1 , p 2 , φ 1 , φ 2 ) to a new set of canonical coordinates
(P
(1)
1 , P
(1)
2 , ,
(1)
1 , ,
(1)
2 ). We write the generating function in the form
S(P
(1)
1 , P
(1)
2 , φ 1 , φ 2 ) =
2
i=1
(P
(1)
i + ξ i )φ i + X(φ 1 , φ 2 )
+
2
i=1
P
(1)
i Y i (φ 1 , φ 2 ),
(2.68)