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I am trying to study the Euler-Lagrange equation. Then I reach this theorem: enter image description here

I check Wikipedia and it seems to say the same thing. Here's what confuse me:

If I understand this theorem correctly, if $y^*$ is a stationary of the functional $F(y)$, then $y^*$ satisfy the Euler-Lagrange equation. But if $y^*$ satisfy the Euler-Lagrange equation, $y^*$ is not necessarily a stationary of the functional $F(y)$, isn't it? Yet in all Physics from how I learned it, they always attempt the Euler-Lagrange equation to derive a differential equation, and solve for the trajectory $y^*$ (like the Brachistochrone problem, or perhaps even Lagrangian and Hamiltonian mechanics), then assume it minimizes $F(y)$

I couldn't find a theorem that stated the vice versa. Could someone enlighten me on this please? Thank you! :D

Source of the picture: https://wiki.math.ntnu.no/_media/tma4180/2015v/calcvar.pdf

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As is usual in optimization, stationarity does not imply optimality (other than in a few special cases, e.g. convexity of the functional). The claim is that if $y^*$ is a minimizer of $L[\cdot]$ with appropriate boundary conditions, then $y^*$ satisfies the Euler-Lagrange equations.

By definition, any $y$ which satisfies the Euler-Lagrange (EL) equation is a stationary point, but it is not necessarily an optimum. So, in some sense, there could be many $y$ which satisfy the EL equation, but often there is only one—this makes the EL equation an extraordinarily useful tool in analyzing functionals.

There is a partial converse to this theorem, which is (and here's the hammer!): if your functional $L[\cdot]$ is coercive (in other words, if $L[\phi] \to +\infty$ as each of $\|\phi\|_2, \|\phi'\|_2 \to \infty$, and $L[\cdot]$ is bounded from below), and there is only one stationary point, then this point is an (in fact, the) optimal point.

In other words if $L[\cdot]$ satisfies the above, then the solution $y^*$ to the EL equation with appropriate boundary conditions is the unique, optimal solution to the problem in question. A particularly simple case of such a functional problem is the functional emerging from the shortest path problem!

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