Take a dynamical map x
M(x)
, with x=(p,q), which is
not integrable: there is no constant of motion F such that
| (1) | F(M(x)) = F(x). |
Assume that the map is almost integrable, in the sense that
| (2) | M(x) = R(x) + ε S(x), |
where x
R(x)
is integrable and ε is infinitesimal.
Now you may want to define an approximate constant Fn of motion for your map, which is constant only up to a power n+1 in ε. That is, we want:
| (3) | Fn (M(x)) = Fn(x) + O(εn+1). |
An iterative process leads us to
| (4) | Fn(x) = f0(x) + ε f1(x) + ... + εnfn(x) |
with a relation of the kind
| (5) | fn(R(x)) = | d fn-1 dx |
(R(x))S(x) + | 1 2! | d2fn-2 dx2 |
(R(x))S(x)2 + | ... + | 1 n! | dnf0 dxn |
(R(x))S(x)n |
Then the series (3) does not converge uniformly. Indeed if it was the case it would define a constant of motion and it would mean that M is an integrable map, which has been discarded.
On the other hand, if you fix n and make ε tend to 0 then relation (3) holds so we have indeed an approximate constant of motion.
Practically it means that for a given ε there is an optimal n for which the series (4) is an approximate constant of motion.