The natural quantum Lagrangians which appear in Galilei general relativistic quantum mechanics are classified by using methods of gauge–natural bundles and natural operators. Namely, all natural quantum Lagrangians are multiples of the canonical scaled volume form, where multiplicative factors are invariant functions. A base of these invariant functions is described: the base is constituted by three functions and the unique non trivial function in the base is just the function corresponding to the natural quantum lagrangian considered by Jadczyk and Modugno.
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.