Intuition behind Euler Lagrange equation in economics When being exposed to the concept of the Euler Lagrange equation as a mathematical concept, many ideas from the physical sciences are used to explain its relevance in terms of choice of shortest path, brachistochrone problem which are great for educational purposes however in the context of economics it is difficult.
The paper that comes to mind which is explict about the Euler Lagrange condition in terms of solving is Grossman 1972 (the famous grossman model of demand for health). The continuous time variant of the model (on page 28-29 of the PDF) has the following condition associated with it (notation changed for simplicity from time denoted by $i$ to being denoted by $t$) we get:
$$\frac{\partial Q}{\partial H(t)}-\frac{\partial}{\partial t}\frac{\partial Q}{\partial \dot{H}(t)}=0$$
or
$$\frac{\partial Q}{\partial H(t)}=\frac{\partial}{\partial t}\frac{\partial Q}{\partial \dot{H}(t)}$$
What is the economic interpretation of this result?
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.