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