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The Functional Bethe Ansatz (FBA) proposed by Sklyanin is a method whichgives separation variables for systems for which an $R$-matrix is known.Previously the FBA was only known for $SL(2)$ and $SL(3)$ (and associated)$R$-matrices. In this paper I advance Sklyanin's program by giving the FBA forcertain systems with $SL(N)$ $R$-matrices. This is achieved by constructingrational functions $\A(u)$ and $\B(u)$ of the matrix elements of $T(u)$, sothat, in the generic case, the zeros $x_i$ of $\B(u)$ are the separationcoordinates and the $P_i=\A(x_i)$ provide their conjugate momenta. The methodis illustrated with the magnetic chain and the Gaudin model, and its widerapplicability is discussed.Comment: 14pp LaTex,DAMTP 94-1

Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain