  In classical mechanics, external constraints on the dynamical variables can
be easily implemented within the Lagrangian formulation. Conversely, the
extension of this idea to the quantum realm, which dates back to Dirac, has
proven notoriously difficult due to the noncommutativity of observables.
Motivated by recent progress in the experimental control of quantum systems, we
propose a framework for the implementation of quantum constraints based on the
idea of work protocols, which are dynamically engineered to enforce the
constraints. As a proof of principle, we consider the dynamical mean-field
approach of the many-body quantum spherical model, which takes the form of a
quantum harmonic oscillator plus constraints on the first and second moments of
one of its quadratures. The constraints of the model are implemented by the
combination of two work protocols, coupling together the first and second
moments of the quadrature operators. We find that such constraints affect the
equations of motion in a highly nontrivial way, inducing nonlinear behavior and
even classical chaos. Interestingly, Gaussianity is preserved at all times. A
discussion concerning the robustness of this approach to possible experimental
errors is also presented.
