Development of canonical transformations from Hamilton’s principle

It is shown that the variations δqj and δQi cannot both be simultaneously zero at the limits of integration in application of Hamilton’s principle to the development of the theory of canonical transformations. This means that δF1 does not vanish. However, the derivation requires only that the δqj be zero at the end points. This lack of symmetry between the qj and Qi in the formulaion is noted with respect to the Lagrangian and the fundamental commutator relations of quantum mechanics. Further, what is meant by an independent variation of pj and qj (or Qi and Pi) in the formalism is discussed in the use of Hamilton’s principle in the derivation of Hamilton’s equations of motion.