Svetlin Georgiev

Abstract:

As it is well known, Isaac Newton had to develop the differential calculus, (jointly with Gottfried Leibniz), with particular reference to the historical definition of velocities as the time derivative of the coordinates, v = dr=dt, in order to write his celebrated equation ma = F (t;r;v), where a = dv=dt is the acceleration and F (t;r;v) is the Newtonian force acting on the mass m. Being local, the dierential calculus solely admitted the characterization of massive points. The differential calculus and the notion of massive points were adopted by Galileo Galilei and Albert Einstein for the formulation of their relativity, thus acquiring a fundamental role in 20th century sciences.

In his Ph. D. thesis of 1966 at the University of Turin, Italy, the Italian-American scientist Ruggero Maria Santilli pointed out that Newtonian forces are the most widely known in dynamics, including action-at-a-distance forces derivable derivable from a potential, thus representable with a Hamiltonian, and other forces that are not derivable from a potential or a Hamiltonian, since they are contact dissipative and non-conservative forces caused by the motion of the mass  m within a physical medium. Santilli pointed out that, due to their lack of dimensions, massive points can solely experience action-at-a-distance Hamiltonian forces.

On this ground, Santilli initiated a long scientic journey for the generalization of Newton's equation into a form permitting the representation of the actual extended character of massive bodies whenever moving within physical media, as a condition to admit non-Hamiltonian forces. Being a theoretical physicist, Santilli had a number of severe physical conditions for the needed representation. One of them was the need for a representation of extended bodies and their non-Hamiltonian forces to be invariant over time as a condition to predict the same numerical values under the same conditions but at different times.

The main purpose in this lecture is to represent some recent researches of Santilli iso-mathematics in the area of the plane geometry. This lecture is devoted to the iso-plane geometry. It summarizes the most recent contributions in this area. Straight iso-lines are introduced. Iso-angle between two iso-vectors is dened. They are introduced iso-lines and they are deducted the main equations of isolines. They are given criteria for iso-perpendicularity and iso-parallel of iso-lines. Iso-reflections, iso-rotations, iso-translations and iso-glide iso-re
flections are introduced. We define iso-circles and they are given the iso- parametric iso-representations of the iso-circles. We introduce iso-ellipse, iso-parabola and iso-hyperbola and they are given some of their basic properties. The lecture is provided with suitable examples.