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1. The problem statement, all variables and given/known data

Understand most of the derivation of the E-L just fine, but am confused about the fact that we can somehow Taylor expand ##L## in this way:

$$ L\bigg[ y+\alpha\eta(x),y'+\alpha \eta^{'}(x),x\bigg] = L \bigg[ y, y',x\bigg] + \frac{\partial L}{\partial y} \alpha \eta(x) + \frac{\partial L}{\partial y'} \alpha \eta^{'} + O^{2}(\alpha \eta) $$

I'm really uncomfortable with the idea that you can treat whole functions as independent of each other (when they are clearly not). I've looked up a lot of derivations online & from textbooks and none of them seem to bother to explain this. Am I missing something obvious? Surely the fact that ## y=f(x,t,\alpha )## should be taken into account somewhere?

Thanks in advance for the help!

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# Euler Lagrange Derivation (Taylor Series)

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