  Considered is the Schr\"odinger equation in a finite-dimensional space as an
equation of mathematical physics derivable from the variational principle and
treatable in terms of the Lagrange-Hamilton formalism. It provides an
interesting example of "mechanics" with singular Lagrangians, effectively
treatable within the framework of Dirac formalism. We discuss also some
modified "Schr\"odinger" equations involving second-order time derivatives and
introduce a kind of non-direct, non-perturbative, geometrically-motivated
nonlinearity based on making the scalar product a dynamical quantity. There are
some reasons to expect that this might be a new way of describing open
dynamical systems and explaining some quantum "paradoxes".
