Hamiltonian Systems on Manifolds of Second-order Tensors *

Considered is the Schrödinger 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 " Schrödinger " 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 ". 1 Finite-level nonlinear Schrödinger equation in Lagrange-Hamilton description Following the jargon used by laser specialists and those working with the quantum dynamics of mutually interacting spins, we use the term " finite-level quantum system " for such a one the " Hilbert space " of which is finite-dimensional, so it may be identified with C n , when some basis is fixed. However, we shall avoid * The article is going to be published in Reports on Mathematical Physics.