  We study the structure of a simple dynamic optimization problem consisting of
one state and one control variable, from a physicist's point of view. By using
an analogy to a physical model, we study this system in the classical and
quantum frameworks. Classically, the dynamic optimization problem is equivalent
to a classical mechanics constrained system, so we must use the Dirac method to
analyze it in a correct way. We find that there are two second-class
constraints in the model: one fix the momenta associated with the control
variables, and the other is a reminder of the optimal control law. The dynamic
evolution of this constrained system is given by the Dirac's bracket of the
canonical variables with the Hamiltonian. This dynamic results to be identical
to the unconstrained one given by the Pontryagin equations, which are the
correct classical equations of motion for our physical optimization problem. In
the same Pontryagin scheme, by imposing a closed-loop $\lambda$-strategy, the
optimality condition for the action gives a consistency relation, which is
associated to the Hamilton-Jacobi-Bellman equation of the dynamic programming
method. A similar result is achieved by quantizing the classical model. By
setting the wave function $\Psi(x,t) = e^{iS(x,t)}$ in the quantum
Schr\"odinger equation, a non-linear partial equation is obtained for the $S$
function. For the right-hand side quantization, this is the
Hamilton-Jacobi-Bellman equation, when $S(x,t)$ is identified with the optimal
value function. Thus, the Hamilton-Jacobi-Bellman equation in Bellman's maximum
principle, can be interpreted as the quantum approach of the optimization
problem.
