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From what I understand I can use the Euler-Lagrange equation to find the function ( Let us call L. ) where L can be the function as stated in the action formula.

But how difficult is it to actually derive the stationary action formula itself or is it the case that once I have he function I can always construct the action formula from first principles ?

Qmechanic
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    The Euler-Lagrange equation doesn't tell you what L is. It uses L that is already defined to find the function that makes the action stationary. Also, do you mean to ask how to use the EL equation to show it makes the action stationary? – BioPhysicist Aug 11 '18 at 20:43
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    Possible duplicates: https://physics.stackexchange.com/q/20298/2451 and links therein. – Qmechanic Aug 11 '18 at 20:55
  • Aaron Stevens ...Oh ...so Euler-Lagrange finds the function that will make the action stationary with the given function. You may have cleared this up for me....with one detail. Can we consider the EL to be both sufficient and necessary for the stationary function of L ? –  Aug 11 '18 at 22:51
  • Most references say the EL is a necessary condition for the extrema to ...I don't see how that helps since EL can be satisfied but that in and of itself does not prove the extrema are there. You would need a sufficient condition not a necessary condition. If A then B means that B is necessary if A is true.. –  Aug 12 '18 at 00:53
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    Just think about maxima and minima from basic calculus. If the first derivative of a function is equal to $0$ at some point, it does not necessarily mean the function is at a maximum or minimum. However, at maxima and minima the first derivative is equal to $0$. The same idea is present here. Also, if you are replying to be specifically in your comments, make sure to tag me (@aaronstevens) so I can see your reply. – BioPhysicist Aug 12 '18 at 06:02

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