  Linear dynamics restricted to invariant submanifolds generally gives rise to
nonlinear dynamics. Submanifolds in the quantum framework may emerge for
several reasons: one could be interested in specific properties possessed by a
given family of states, either as a consequence of experimental constraints or
inside an approximation scheme. In this work we investigate such issues in
connection with a one parameter group $\phi_t$ of transformations on a Hilbert
space, $\mathcal{H}$, defining the unitary evolutions of a chosen quantum
system. Two procedures will be presented: the first one consists in the
restriction of the vector field associated with the Schr\"{o}dinger equation to
a submanifold invariant under the flow $\phi_t$. The second one makes use of
the Lagrangian formalism and can be extended also to non-invariant
submanifolds, even if in such a case the resulting dynamics is only an
approximation of the flow $\phi_t$. Such a result, therefore, should be
conceived as a generalization of the variational method already employed for
stationary problems.
