I am trying to get a better understanding of the Lagrangian. From what I know, we say that each trajectory in physics must be a path that is at a minimum, which means that is satisfies the Euler-Lagrange equations. Thus, we define the Lagrangian, which is exactly this function.
Now, considering a non-relativistic newtonian problem, it is stated that the Lagrangian is the function $$ L = T - U $$ How does this function satisfy the Euler-Lagrange equations, and more importantly, how is it derived?
