# Lagrange Multipliers Versus Generalized Coordinates

When forced to explain to someone why one could either set up a general Lagrangian & then incorporate constraints using Lagrange multipliers, as opposed to just setting up a Lagrangian with generalized coordinate built in from the start I found I couldn't do it - I don't actually know why one can use either method other than that it apparently works. Is there some theorem or some substitution that says either method is valid, or is this just stunningly obvious & I'm missing it?

I've re-checked one of my video courses where the guy solves a problem using three different methods but never mentions why they're equivalent, checked both mechanics & calculus of variations books to find an explanation & checked posts on this forum as well as other forums but seem to have missed it, thus I'd really appreciate any commentary & references from you guys on this - thanks for reading!

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Extremizing the action $S_\textrm{full}$ with Lagrange multipliers $$\delta S_\textrm{full} = 0,\quad S_\textrm{full} = S_\textrm{orig}+\sum\int \lambda (g(x^i)-c)$$ may be seen to imply $g(x^i)=c$ – that's the derivative of the full action with respect to the Lagrange multiplier(s) $\lambda$. Because $\delta S_\textrm{full} = 0$ implies $g(x^i)=c$, among other things, we may assume this relationship while extremizing $S_\textrm{full}$ on the subspace of the configuration space that obeys the conditions $g(x^i)=c$. But on this submanifold, $S_\textrm{full}=S_\textrm{orig}$, so the two extremization conditions are equivalent.