  Unfortunately, the Hamiltonian mechanics of degenerate Lagrangian systems is
usually presented as a mere recipe of Dirac, with no explanation as to how it
works. Then it comes to discussing conjectures of whether all primary
constraints correspond to gauge symmetries, and it goes all the way to
absolutely wrong claims such as the statement that electrodynamics or gravity
have only two physical components each, with others being spurious. One has to
be very careful because non-dynamical, or constrained, does not mean
unphysical. I give a pedagogical introduction to the degenerate Hamiltonian
systems, showing both very simple mechanical examples and general arguments
about how it works. For the familiar field theory models, I explain why the
gauge freedom there "hits twice" in the sense of producing twice as many
first-class constraints as gauge symmetries, and why primary, and only primary,
constraints should be put into the total Hamiltonian.
