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Hamilton's differential equations have an origin. They arise neither from nothingness nor from individual genius, but from the painful "work of the concept". They appear as a curiosity in the calculations of J. L. Lagrange and acquire their theoretical formulation in the work of C. G. J. Jacobi. Among other things, this book deals with that history and how mathematical theories develop. The axis of the development of the text is the problem of hamiltonization, which from being very specialized must become central in the construction of the mathematical theory of mechanics.