We consider {\it local} balances of momentum and angular momentum for the
incompressible Navier-Stokes equations. First, we formulate new weak forms of
the physical balances (conservation laws) of these quantities, and prove they
are equivalent to the usual conservation law formulations. We then show that
continuous Galerkin discretizations of the Navier-Stokes equations using the
EMAC form of the nonlinearity preserve discrete analogues of the weak form
conservation laws, both in the Eulerian formulation and the Lagrangian
formulation (which are not equivalent after discretizations). Numerical tests
illustrate the new theory