Hamiltonian Systems Inspired by the Schrödinger Equation - Kovalchuk, V.; Slawianowski, J.J. (Symmetry, Integrability and Geometry: Methods and Applications, 2008)
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 Schrödinger equations are obtained.