vlcek
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11/30/06


NEW COORDINATE SYSTEMS
Posted:
Aug 9, 2011 3:45 PM


THE NEW COORDINATE SYSTEMS Take the minimal number of identical particles with a globelike form and forming the nearest organized configuration. This configuration is a disfenoid at the vertices with four particles (the  particle has 4 nucleons). The origin of our new coordinate system is put into the center of gravity of the configuration. This origin and the centres of the particles determine the semilinessemiaxes of the coordinate system. This coordinate system divides the space into four quartespaces. The pairs of semilines (s,t), (s,u), (s,v), (t, u), (t,v), (u,v) determine angles For angle it precisely holds:
To imagine better the coordinate system (s,t,u,v), we can use the cube. The centre of the cube is the center of gravity of the disfenoid and also the origin of the coordinate system (see fig. 1). In order to facilitate the transformation to the cartesian coordinate system, this will be somewhat rearranged: semiaxes x,y,z will have the same marks, semiaxes (x),(y),(z) will be marked and so cartesian coordinate system (x,y,z) in the new marking will be revealed as a system ( ) formed by semiaxes . These, regarding to the coordinate system (s,t,u,v), will be determined as follows: semiaxis x is the symmetral of the angle (see fig. 2) semiaxis y is the symmetral of the angle in "  plane" (t,u) semiaxis z is the symmetral of the angle in "  plane" (u,v) semiaxis is the symmetral of the angle in "  plane" (t,v) semiaxis is the symmetral of the angle in "  plane" (s,v) semiaxis is the symmetral of the angle in "  plane" (s,t).
Fig. 1. The coordinate system (s,t,u,v)
Fig. 2. The semi  axis x is the symmetral of the angle After drawing both coordinate system we will achieve fig. 3. The cartesian coordinate system divides the space into 8 octants: . The trinities of "  planes" determine four equal quarterspaces: (s,t,u), (s,t,v), (s,u,v), (t,u,v). It is impossible to divide the space into equal parts using less than 4 semiaxes. It means that these quarterspaces are the largest possible parts of the space formed by the minimal number of semiaxes.
Fig. 3. Both coordinate system (s,t,u,v) and ( ) The values of coordinates will be read in two ways: a) The straight lines placed from an arbitrary point parallel to the axes s,t,u,v determine coordinates s,t,u,v, fig. 4. Zero in the contained coordinate means that the point is placed in the quarter space determined by coordinates other than zero. See the following transformation equation between (s,t,u,v) and ( ): (s,t,u)
(s,u,v)
(s,t,v)
(t,u,v)
Fig. 4. The coordinates (s,t,u,v) (s,t,u): (s,u,v):
(s,t,v): (t,u,v):
The distance between two points (s1,t1,u1,v1) and (s2,t2,u2,v2) is determined as follows:
b) The planes placed from an arbitrary point perpendicular to the axis s,t,u,v determine coordinates s*,t*,u*,v*, see fig. 5.
Fig. 5. The coordinates s*,t*,u*,v* See the following transformation equations between s*,t*,u*,v*, and :
(s*,t*,u*): (s*,u*,v*):
(s*,t*,v*): (t*,u*,v*):
Quadrate of distance between two points (s1*,t1*, u1*,v1*) and (s2*,t2*, u2*,v2*) is determined by this equation:
See the following transformation equation between (s,t,u,v) and (s*,t*,u*,v*): (s,t,u): (s,t,v):
(s,u,v): (t,u,v):
(s,t,u): (s,t,v):
(s,u,v): (t,u,v):
Rotation around axis x,y,z  the angle of rotation is  are invariant. They perform the disfenoid into equivalent positions. The rotations around the axis s,t,u,v  the angle of rotation is  are the invariant ones. E,s,s1,t,t1,x,u,u1,y,v,v1,z form the group of rotation, see Tab. 1. Table 1. columns  Acts as the first, rows  Acts as the second E s s1 t t1 u u1 v v1 x y z E E s s1 t t1 u u1 v v1 x y z s1 s1 E s z u y v x t u1 t1 v1 s s s1 E v1 y t1 x u1 z v u t t1 t1 z v E t x s y u v1 s1 u1 t t u1 y t1 E v1 z s1 x u v s u1 u1 y t x v E u z s s1 v1 t1 u u v1 x s1 z u1 E t1 y t s v v1 v1 x u y s z t E v t1 u1 s1 v v t1 z u1 x s1 y v1 E s t u x x u v1 v u1 s t1 t s1 E z y y y t u1 s v1 v s1 u t1 z E x z z v t1 u s1 t v1 s u1 y x E
REFERENCES [1] MAYER, M. G.: Phys. Rev. 74, 235, (1948) [2] FEJES Tóth, L.: Am. Math. Month. 56, 330 (1949) [3] HABICHT W., van der WAERDEN, B. L.: Math. Ann. 123, 223 (1951) [4] WHYTE, L. L.: Am. Math. Month. 59, 606 (1952) [5] LEECH, J.: Math. Gaz. 41, 81 (1957) [6] BEREZIN, A. A.: Nature (London) 317, 208 (1985) [7] BEREZIN, A. A.: J. Math. Phys. 27(6), 1533 (1986)
See you please L. Vlcek : New Trends in Physics, Slovak Academic Press, Bratislava 1996 ISBN 8085665646. Presentation on European Phys. Soc. 10th Gen. Conf. Trends in Physics ( EPS 10) Sevilla , E
http://www.trendsinphysics.info/

