Suppose we have an integrable system consisting of a $2n$-dimensional phase space $M$ together with $n$ independent functions $f_{1\leq j \leq n }$ in involution. Suppose the level set
$$M_f = \{ (p,q)\in M | f_k(p,q)= c_k \}, \text{where} \ c_k = \text{const.},$$
is compact so that it is diffeomorphic to a torus. The action variables are then defined as the circle integrals over the $n$ independent circles on the torus:
$$ I_k = \oint_{C_k} \sum_k p_k dq_k . $$
The apparent question is then, why do we bother to define a new set of variables, i.e. the action variables in this way? Why not just take the $f$'s as the action variable? Anyway, they are already in involution.
