Concepts and Principles

After presenting the basic concepts to start the study of mechanics, such as time, space, particle and mass, a necessary mathematical review of implicit and explicit dependence of variables and variational calculus is made. Then, Hamilton’s principle is introduced as the common origin of all phenomena in classical mechanics. The discovery of the form of the Lagrangian function and its dependence on dynamic variables, to be used with Hamilton’s principle, is presented as the true theoretical challenge, exemplifying the inductive character of physical theories. Several examples of Lagrangian functions different of the “standard” form, \(L=T - V\) , are discussed for a particle free of constraints, that is, prior to the introduction of generalized coordinates and the \(L=T - V\) form. Obtaining Newton laws from Hamilton’s principle and symmetries of nature, reinforces the basic idea of the chapter, that is, the prevalence of Hamilton’s principle, which works even in cases where Newton’s laws do not, as in special relativity.