5. Euler's equation for the simplest problem.
Theorem 1: If a functiony is extremal to the function
wherey belongs to C2[a,b] and
then y must fulfill the Euler equation
Remark: The Euler equation corresponds to the condition
in extreme value problems and is an ordinary differential equation that in general is nonlinear. By using the chain rule, we see that it can be expanded as
The function L=L(x,y,y‘) is called the Lagrangian of the problem. In our previous problems problems 4, 5 and 6 the Lagrangians were