This would be, as explained above, the integral of this equation over time.
Here, t1 and t2 are the initial and final time positions, and L(x,x') is our LAGRANGIAN, which is exactly our Energy Equation, but with another name.
∂L/∂xi - d/dt ( ∂L/∂xi' ) = 0
And then use the mathematical technique of the Lagrangian Multipliers (which I will not explain here), in order to simplify our Action Integral into something we can operate with, i.e. a second order differential equation.
And this is it. The basis, the cornerstone of Lagrangian Dynamics, demystified.
Once these concepts are comprehended, it is relatively easy to toddle around all the rest of this vast world...now at least we have legs!