  The Hamiltonian formulation plays the essential role in constructing the
framework of modern physics. In this paper, a new form of canonical equations
of Hamilton with the complete symmetry is obtained, which are valid not only
for the first-order differential system, but also for the second-order
differential system. The conventional form of the canonical equations without
the symmetry [Goldstein et al., Classical Mechanics, 3rd ed, Addison-Wesley,
2001] are only for the second-order differential system. It is pointed out for
the first time that the number of the canonical equations for the first-order
differential system is half of that for the second-order differential system.
The nonlinear Schr\"{o}dinger equation, a universal first-order differential
system, can be expressed with the new canonical equations in a consistent way.

Classical Physics;Exactly Solvable and Integrable Systems