82
3 Area-Preserving Maps
Fig. 3.22 The shaded regions show the area pumped across the line x = 0 in the positive p
direction by the rational approximate with winding number ω =
0
1 after one iteration of the
standard map: (a) K = 0.0, (b) K = 0.47, (c) K = 0.97, (d) K = 1.97
F (x n , x n+1 ) =
1
2
(x n − x n+1 )
2
+
K
(2π) 2 cos(2πx n ).
(3.64)
If we let p n+1 = x n+1 − x n , it is easy to see that the standard map can be written in
the Newtonian form as
x n+1 + x n−1 − 2x n = −
K
2π
sin(2πx n )
(3.65)
(Percival 1979). Equation (3.65) can be obtained by extremizing W n−1,n =
F (x n−1 , x n ) + F (x n , x n+1 ) with respect to the intermediate variable, x n (see
Sect. 3.2).
Let us now compute the area (or action) pumped across our original line in
Fig. 3.22a. The shaded area is
A shaded =
1
1
2
p 1 (t)
dx 1
dt
dt −
1
1
2
p 0 (t)
dx 0
dt
dt.
(3.66)