When deriving stationary points of a function defined by a 1-D integral (think lagranian mechanics, Fermat's priniciple, geodesics, etc) and arriving at the Euler Lagrange equation, there seems to me to be an unjustified assumption in the derivation. The derivations I have seen start with something along the following lines: assume some function x(t) is the function we are looking for, let x'(t) = x(t) + η(t) be a nearby path... The derivation will then go on to show the conditions for the original function x(t), namely that the function satisfy the Euler Lagrange equation.(adsbygoogle = window.adsbygoogle || []).push({});

It seems a little odd that we assume, without proof, that this function exists and then sort out its properties. How do we know such a function exists? Does it always exist? Are there conditions on this? Isn't it a little shady to be discussing properties of something if we havent proved yet that it exists?

On the other hand, once we complete the derivation, it seems clear to me that a function which satisfies the Euler Lagrange equation will be a stationary function. I think.

I'm still left feeling uncomfortable however about this. Is there some outside proof which shows that this function must exist?

I shoud give the caveat that I have only seen this derivation in physics books, I dont own any math books on calculus of variations

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# Questioning an assumption in calculus of variations

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