Three Simultaneous Superimposed Rotating System

Abstract

In inertial system, co-ordinate transformation from one frame to another is possible by using Lorentz transformation matrix. But in non-inertial or rotating system it is not applicable by using Lorentz transformation matrix. In this paper, co-ordinate transformation from one frame to another in three simultaneous superimposed rotating systems has been introduced. This also leads to assume a picture of space-time geometry of same system. Keywords: non-inertial system; co-ordinate transformation; space-time geometry THREE SIMULTANEOUS SUPERIMPOSED ROTATIONAL MOTIONAccording to [1,2] A particle or an event may possess two simultaneous superimposed motions (i.e. either linear or rotational). Similarly a particle or an event may possess three simultaneous superimposed rotational motions. For clarity of three simultaneous superimposed rotational motions as well as three simultaneous superimposed spins, following [3], it is stated that frames S and S 1 have both their X axes aligned and S 1 is moving at an angular velocity 1  about X 1 axis as observed by S. The frame S 1 has another co-ordinate reference frame S 2 , where X 2 axis of S 2 , are rotated by an angle  counter clockwise with respect to S 1 on X 1 Y 1 plane. Frames S 2 and S 3 have both their X axes aligned and S 3 is moving at an angular velocity 2  about X 3 axis as observed by S 2 . Similarly S 3 has another co-ordinate reference frame S 4 , where X 3 axis of S 3 , are rotated by an angle  counter clockwise with respect to S 3 on X 3 Y 3 plane. Frames S 4 and S 5 have both their X axes aligned and S 5 is moving at an angular velocity 3  about X 5 axis as observed by S 4 .For the case when the origin of frames are same with respect to S and the particle be at the origin of S 3 , then it possesses three simultaneous superimposed spins with respect to frame S. Coordinate transformation matrix from S to S 5 would be

Three Simultaneous Superimposed Rotating System

“…Similarly a particle or an event may possess three simultaneous superimposed rotational motions. For clarity of three simultaneous superimposed rotational motions as well as three simultaneous superimposed spins, following [3], it is stated that frames S and S 1 have both their X axes aligned and S 1 is moving at an angular velocity 1  about X 1 axis as observed by S. The frame S 1 has another co-ordinate reference frame S 2 , where X 2 axis of S 2 , are rotated by an angle  counter clockwise with respect to S 1 on X 1 Y 1 plane. Frames S 2 and S 3 have both their X axes aligned and S 3 is moving at an angular velocity 2  about X 3 axis as observed by S 2 .…”