Introduction of an Hamiltonian Function and a Canonical Representation in view of a Possible Optimal Control of a Cellular Colony

In this paper the study of the behaviour of a cellular colony in controlled growth is illustrated. In particular we deal with the following tasks: how to introduce the Hamiltonian function in an analytic nonlinear evolutionary process and how to obtain a canonical representation of the process by means of two sequences of equations, similarly to what is done in finite optimal processes theory. In this path we are forced to introduce "adjoint variables", namely "generalized momenta", to be considered together with the "positional variables". If they may have a plain physical meaning, we are legitimate not only to use the language of classical mechanics, but also to investigate other analogies in order to supply with their consequences the study of similar processes. Relevant in this framework is a biological problem: the study concerns a birth and death stochastic process whose canonical representation is allowed by generalized momenta which have a clear interpretation. Its pre-eminence in economy of our studies stands in possibility that it might be the starting point for the optimal therapeutic conduct in fight against tumors.