Intuition behind Euler Lagrange equation in economics
The Euler Lagrange Equation in words in the context of physical capital accumulation states: Any possible accumulation that may occur at the optimum is brought back to a steady state in the next instant due to the law of motion.
This equation communicates the physical properties of our optimum meaning that once we are there no possible improvement from capital accumulation is possible.
This logic carries over for the our case in the context of the continuous time version of the Grossman model this time with health capital.