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\begin{equation} f(x,x',y,y',t) \end{equation} and we are trying to minimise this subject to a constraint of

\begin{equation} g(x,x',y,y',t) \end{equation}

Would we simply have a set of two euler lagrange equations for each dependent variable, here we have x and y.

Would we insert f(x,x',y,y',t)-Ag(x,x',y,y',t) into both equations, where A is a constant? Or would each equation require a different constant in front of the constraint term g(x,x',y,y',t)??