52
3 Area-Preserving Maps
p n = −
∂F
∂φ n
φ n+1
, and p n+1 =
∂F
∂φ n+1
φ n
.
(3.7)
The area-preserving property of the map makes the differential,
dF = ρ n+1 dφ n+1 − ρ n dφ n ,
(3.8)
exact. This can be seen as follows. If we rearrange the partial derivatives, we can
write the Jacobian in Eq. (3.6) as
J
ρ n+1 φ n+1
ρ n φ n
= −
∂ρ n+1
∂φ n
φ n+1
∂φ n+1
∂ρ n
φ n
=
∂ 2 F
∂φ n ∂φ n+1
∂ 2 F
∂φ n+1 ∂φ n
= 1.
(3.9)
Area-preservation requires
∂ 2 F
∂φ n ∂φ n+1
=
∂ 2 F
∂φ n+1 ∂φ n
, which is also the condition
for exactness of the differential dF .
The twist maps most commonly used have a simple form:
ρ n+1 = ρ n + (φ n ),
(3.10)
φ n+1 = φ n + ρ n + (φ n ).
(3.11)
The generating function for such maps is of the form
F (φ n , φ n+1 ) =
1
2
(φ n − φ n+1 )
2
+ V (φ n ),
(3.12)
where f (φ n ) =
∂V
∂φ n
.
The generating function, F (φ n , φ n+1 ), is a Lagrangian, and some properties of
the map may be derived from a stationary action principle.
Action Principles for Discrete Maps (MacKay et al. 1984)
I. If φ n−1 , φ n , φ n+1 are three successive points of an orbit, then
∂
∂φ n
(F (φ n−1 , φ n ) + F (φ n , φ n+1 )) = 0
and conversely.
II. Let {φ n } denote a sequence of points with initial and final values φ r and φ s , respectively,
specified. This sequence defines a segment of a T orbit if and only if the action sum
W r,s =
s−1
n=r
F (φ n , φ n+1 )