  We show how engineered classical noise can be used to generate constrained
Hamiltonian dynamics in atomic quantum simulators of many-body systems, taking
advantage of the continuous Zeno effect. After discussing the general
theoretical framework, we focus on applications in the context of lattice gauge
theories, where imposing exotic, quasi-local constraints is usually
challenging. We demonstrate the effectiveness of the scheme for both Abelian
and non-Abelian gauge theories, and discuss how engineering dissipative
constraints substitutes complicated, non-local interaction patterns by global
coupling to laser fields.
