Classical and quantum Hamilton formalisms for the mechanics of the Bernoulli oscillators I : Classical framework

In a few previous papers, we discussed the fundamentals of the so-called Bernoulli oscillators physics. The Bernoulli oscillators are classical entities whose behavior is influenced by a ''hidden'' degree of freedom (HDF), in its turn excited by a quantum vacuum action. Within assessed approximations and limits (uni-dimensional motion), we assumed a classical-like interpretation of quantum effects, and displayed the Newtonian motion background subtending - by our proposal - the matter wave physics. In a couple of papers, we give a formal description of the Bernoulli oscillators classical degree of freedom mechanics, by the means of Hamilton-like formalisms. These are, however, different in their conception from the standard known ones: we introduce indeed an extended form for the Hamilton function, called the Bernoulli Hamiltonian, and non-standard forms for the Hamilton equations or procedures. Due to the flexibility of our forms, both a classical framework and a quantum-like one will be shown able to provide a full description of all the cases relevant to us. In the present paper I, the classical framework is discussed. Physical interpretation matching the mentioned formal procedures is provided step-by-step.