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


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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.