Post date: Sep 05, 2018 1:41:40 PM
See [1] for a well-discoursed paper on the topic, and [2] for the original paper by Bellman himself (and communicated by the John Von Neumann!).
While the Bellman equation approaches the functional optimization problem from the multistage-decision-making perspective,
the classical approach of finding the first order necessary condition yields the Euler-Lagrange equation.
Regardless, both the Bellman and Euler-Lagrange equations provide the condition at every point which the unknown function must satisfy to optimize its functional.
And they are shown to be equivalent in the references below.
[1] https://core.ac.uk/download/pdf/81974511.pdf
[2] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC527981/pdf/pnas00731-0009.pdf