  When discussing consequences of symmetries of dynamical systems based on
Noether's first theorem, most standard textbooks on classical or quantum
mechanics present a conclusion stating that a global continuous Lie symmetry
implies the existence of a time independent conserved Noether charge which is
the generator of the action on phase space of that symmetry, and which
necessarily must as well commute with the Hamiltonian. However this need not be
so, nor does that statement do justice to the complete scope and reach of
Noether's first theorem. Rather a much less restrictive statement applies,
namely that the corresponding Noether charge as an observable over phase space
may in fact possess an explicit time dependency, and yet define a constant of
the motion by having a commutator with the Hamiltonian which is nonvanishing,
thus indeed defining a dynamical conserved quantity. Furthermore, and this
certainly within the Hamiltonian formulation, the converse statement is valid
as well, namely that any dynamical constant of motion is necessarily the
Noether charge of some symmetry leaving the system's action invariant up to
some total time derivative contribution. The present contribution revisits
these different points and their consequences, straightaway within the
Hamiltonian formulation which is the most appropriate for such issues. Explicit
illustrations are also provided through three general but simple enough classes
of systems.
