Hamiltonian Systems Inspired by the Schr\"odinger Equation
  Described is n-level quantum system realized in the n-dimensional ''Hilbert'' space H with the scalar product G taken as a dynamical variable. The most general Lagrangian 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 usual, first- and second-order modified Schr\"odinger equations are obtained. 
Vasyl Kovalchuk and Jan Jerzy Slawianowski