In the action-angle formalism, the arguments of Hamilton’s characteristic function are the...

In the action-angle formalism, the arguments of Hamilton’s characteristic function are the original coordinates qk and the action variables Jk. In the case of degeneracy, a subsequent canonical transformation is made to new variables from (wk, Jk), in order to replace the degeneracies by zero frequencies. By considering each Jk a function of the quantities as defined by Eq. (10.126), show that it remains true that