  Noether's calculus of invariant variations yields exact identities from
functional symmetries. The standard application to an action integral allows to
identify conservation laws. Here we rather consider generating functionals,
such as the free energy and the power functional, for equilibrium and driven
many-body systems. Translational and rotational symmetry operations yield
mechanical laws. These global identities express vanishing of total internal
and total external forces and torques. We show that functional differentiation
then leads to hierarchies of local sum rules that interrelate density
correlators as well as static and time direct correlation functions, including
memory. For anisotropic particles, orbital and spin motion become
systematically coupled. The theory allows us to shed new light on the
spatio-temporal coupling of correlations in complex systems. As applications we
consider active Brownian particles, where the theory clarifies the role of
interfacial forces in motility-induced phase separation. For active
sedimentation, the center-of-mass motion is constrained by an internal Noether
sum rule.
