The variation principle has not been well defined in quantum mechanics for a Lagrangian containing a velocity dependent potential. We investigate the method of variation to derive an Euler-Lagrange equation in connection with Noether's theorem when the Lagrangian is invariant under some point transformation. Obtained are the Euler-Lagrange equation retaining the above invariance by means of the variation method and the Hamiltonian with which the canonical equation of motion leads to the Euler-Lagrange equation.
A previous q-number variational method is extended to be applicable to any quantum system for which the classical Lagrangian is given by Lc=½gij\dotqi\dotqj+ui\dotqi-v. The q-number variation is necessary for the formulation to be form-invariant under a general space-time transformation. From the action principle, not only the Euler-Lagrange equation but also the commutation relations up to a constant factor are obtained. Futhermore the form of the quantum Lagrangian is, to some extent, decided in order for solutions to exist for the action principle. It is shown that the first Noether theorem holds and the quantization is consistent with the canonical formalism.