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 $m a = F(t, r, v)$, where $a = dv/dt$ is the acceleration and $F(t, r, v)$ is the Newtonian force acting on the mass $m$. Being local, the differential 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 scientific 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 resulting new calculus, today known as  Santilli IsoDifferential Calculus, or IDC for short, stimulated a further layer of studies that finally signaled the achievement of mathematical and physical maturity. In particular, we note: the isotopies of Euclidean, Minkowskian, Riemannian and symplectic geometries; the  isotopies of classical Hamiltonian mechanics, today known as the  Hamilton-Santilli isomechanics, and the isotopies of quantum mechanics, today known as the  isotopic branch of Hadronic mechanics.

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 defined. They are introduced iso-lines and they are deducted the main equations of iso-lines. They are given criteria for iso-perpendicularity and iso-parallel of iso-lines.  Iso-reflections, iso-rotations, iso-translations and iso-glide iso-reflections 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.