Authors: Jerzy Hanckowiak
Comments: 15 Pages.
Constraints imposed directly on accelerations of the system leading to the relation of constants of motion with appropriate local projectors occurring in the derived equations are considered. In this way a generalization of the Noether's theorem is obtained and constraints are also considered in the phase space.
Category: Mathematical Physics
Authors: Markos Georgallides
Comments: 12 Pages.
This article explains how Geometry and geometrical constructions , [ Spaces , Anti-Spaces , Sub-Spaces ] are able to connect Properties , such as Mass , Energy , and the conservation laws of Physics with only one substance , that of Vector ds , which is ds = AB > 0 → < ∞ with Impulses P A + P B = 0 or ≠ 0 at edge points A , B . Since neutrinos may travel faster than light and also gravity instantly affects , a new explanation is needed . The present
article < The Higgs Boson ( Particle ) and the Euclidean Geometry > describes the motion of one of the infinite Sub-Spaces in [ PNS ] , which are moving in Primary Neutral Space . It has been shown that ,
1.. Primary point A , having not Position and consequently not any Space existing , therefore is the only Space , and to exist at a second point B somewhere else , point A must move at point B , where then A ≡ B . Point B is the Primary Anti-Space which Equilibrium point A [ PNS ] = [ A ≡ B ] . Since Primary point A is the only Space , then on it exists Principle of Virtual Displacements W = ∫ P.ds = 0 , or [ ds . ( PA + P B ) = 0 ] , i.e. for any ds > 0 Impulse P = ( PA + P B ) = 0 . All points may exist with P = 0 → ( PNS ) and also with P ≠ 0 , [ P A + P B = 0 for points in Spaces and Anti - Spaces ] , therefore [ PNS ] is self created , and because at each point may exist also P ≠ 0 , then [ PNS ] is a Field with infinite points which have a ± Charge with P = 0 → P → ∞ . Primary Neutral - Space [ PNS ] exists with the infinite Points N with three Spatial dimensions ( Xo , Yo , Zo ) and the infinite Impulses P , ( P xi , Pyi , Pzi ) , and i = 0 → ∞ .
2.. Since points A , B of [ PNS ] coincide with the infinite Points , of the infinite Spaces , Anti-Spaces and Sub-Spaces of [ PNS] , and since Motion may occur at all Bounded Sub-Spaces , then this Relative motion is happening between all points belonging to [ PNS ] and to those points belonging to the other Sub-Spaces ( A ≡ B ) . The Infinite points in [ PNS ] form infinite Units AiBi = ds , which equilibrium by Primary Anti – Space by an Inner Impulse ( P ) at edges A , B where PiA + Pi B ≠ 0 , and ds = 0 → N = ∞ .
3 .. Monads = Quantum = ds = AB / ( n = ∞ → 0 ) = [ a ± b.i ] = 0 → ∞ create Spaces ( S ) , Anti -Spaces ( A-S ) , and Sub-Spaces ( S-S ) of AB , which Sub-Spaces are Bounded Spaces , Anti-Spaces and Sub Spaces in it , and are not purely spatial because are Complex numbers which exist for all Spaces , since binomial dsⁿ is Complex number also .
4.. Monads ds ( dipole AB ) and according to their position in S , A-S , S-S , make the four types of matter , and the Combination of the four types of matter , creates all gauge magnitudes which is Mass and Energy , and so all types of Particles and Fields in universe .The 15 possible types of matter correspond to the , Visible , Invisible , Real and Imaginary Universes ( Visible and Invisible Dark matter and Energy ) . Dipole AB is composed of the two Elements → the [ Dipole AB = matter ≡ is the communicator] , and the Impulse [ P ] with the Bounded Impulses ( PA , P B ) .
5.. The difference of Impulses dP = P B – PA > = < 0 of Dipole AB (mass) , creates ± charge . For very small ds ( near zero) which happens at the boundaries of Spaces , ± charge may be point charge ( Spaces form open Strings while Sub-Spaces form closed Strings . Cosmic rays is a Quantum field from a , near zero ds Sub-Space ) , and in very large scales ds ,
is of ± charge and of the same Field , ( This because Particles , wave like , do not enter a Space smaller than their wavelength ) .
6.. A ray , say M - Ray , interact on the infinite Monads of [PNS] with infinite velocity and zero frequency . This Motion is Continuous and occurs on Dimensional Bounded Units , ds , and not on Points ( this happens only for very small ds near zero and at boundaries of the Spaces ) which are dimensionless . The difference of Impulse dP = P B-PA on points A , B and on points Pi A , Pi B is ± charge . All particles act as wave ( wave-particle duality ) because of the Total energy conservation law of Pythagoras in 2,3 Dim .
7.. The four Interaction of particles or Fields occur directly or Indirectly .
Category: Mathematical Physics