[AntonioLao]

Is it possible to absolutely transformation one spacetime point to another? Mathematically speaking, the answer is barely or more appropriately next to nothing. Moreover, a special branch of calculus called ‘calculus of variations’ is the technique of dealing with the optimization (finding maxima ad minima) of summing all the infinitesimal contributions. These significant contributions lead to a search for a principle of least action. The study of analytical mechanics found this principle in the form of Lagrange’s equations. However, Lagrange’s equations are consequences from point transformations. This is more applicable to conservative systems. A more general principle applicable to non-conservative systems using contact transformations is Hamilton’s principle.

Hamilton’s principle is also an equivalence principle analogous to the equivalence of gravitational mass and inertial mass. The difference is that it is an equivalence principle for infinitesimal contact transformation between gravitational energy and inertial energy. Generally, it is an equivalence principle between infinitesimal potential energy and infinitesimal kinetic energy or using d’Alembert’s terminology ‘virtual work’ done by scalar forces and vectorial forces. Nevertheless, d’Alembert’s principle operates with non-integrable differential, a certain infinitesimal quantity defined as the equality of infinitesimal kinetic energy and infinitesimal potential energy. It was of greatest theoretical and practical importance that a transformation brings d’Alembert’s ‘vectorial’ kinetic energy into a ‘scalar’ kinetic energy in order for it to be integrable with the potential energy derived from scalar pressure forces.