Hamiltonian systems inspired by the Schrödinger equation

Abstract:
Described is n-leveled quantum system realized in the n-dimensional "Hilbert" space H with the scalar product G taken as a dynamical variable. The most general Langrangian for the wave function and G is considered. Equations of motion and conservation laws are obtained. Special cases for the free evolution of the wave function with fixed G and the pure dynamics of G are calculated. The ordinary and first-order modified Schrödinger equations are obtained.