I asked a question recently on here and got very insightful answers. Now that I understand the proof of the variational method to obtain the EL equation, I have a new question.
The proof that I went through started with the supposition that there is a special path $y(t)$ which minimized the action integral $S$. We then proceeded to create a family of paths which $y(t)$ belongs to, and this family is parametrized by $\alpha$. From design we know that the path which minimizes the action integral must occur at $\alpha = 0$ from the initial supposition. After some integral manipulation, we get the EL equation (which is satisfied by the special path $y(t)$).
My question is this. If we were to go backwards and find some path that satisfies EL, does that have to imply that it gives the minimum of the action?