Derivation of theories: structures of the derived system in terms of those of the original system in classical mechanics

We present the technique of derivation of a theory to obtain an $(n+1)f$-degrees-of-freedom theory from an $f$-degrees-of-freedom theory and show that one can calculate all of the quantities of the derived theory from those of the original one. Specifically, we show that one can use this technique to construct, from an integrable system, other integrable systems with more degrees of freedom.