Divergent series


 

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) + &epsilon S(x),

where x R(x) is integrable and &epsilon 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(&epsilonn+1).

An iterative process leads us to
(4) Fn(x)  =  f0(x) + &epsilon f1(x) + ... + &epsilonnfn(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 &epsilon tend to 0 then relation (3) holds so we have indeed an approximate constant of motion.

Practically it means that for a given &epsilon there is an optimal n for which the series (4) is an approximate constant of motion.