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There is a pdf i found when searching about Lagrangian Multpliers, but i was not able to understand how he derived lambda from two differential equations. If anyone can walk me through it, i would be really grateful. enter image description here

I have marked the equation he derived with a black pen. Thanks.

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  • $\begingroup$ @AccidentalFourierTransform could you explain me how he has done it, i tried all the ways i know but i was not getting the result he got. $\endgroup$ Feb 13, 2016 at 16:33

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The derivation is the following:

$2\lambda x - m\ddot{x} = 0 \rightarrow 2\lambda x^2 = mx\ddot{x}~~~~(1)$

$2\lambda y - m\ddot{y} - mg = 0 \rightarrow 2\lambda y^2 = my\ddot{y} + mgy~~~~(2)$

Now, we sum Eqs. (1) and (2):

$2\lambda (x^2+y^2) = m(x\ddot{x}+y\ddot{y} + mgy)$,

where $x^2+y^2 = l^2$. Then,

$\lambda = \frac{m}{2l^2}(x\ddot{x}+y\ddot{y} + gy)$.

Hope this helps.

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