Reference frame theory coordinate transformation

In the sequel, some general issues concerning AC-machines modeling, mainly change of variables through coordinate transformation (reference-frame theory, see [8]) are discussed. This is followed by the presentation of machine schematics in machine variables and equivalent circuits in transformed variables of both the synchronous and the induction machines, under detailed modeling assumptions. This is accompanied by the corresponding BG models of both machines in transformed variables. Finally, simplified models of the induction motor usually encountered in control system applications are addressed. 

The set of transformations of the spacetime coordinates that project the laws of electrodynamics from any observer s reference frame to any other (continuously connected) inertial frame such that the laws remain unchanged is the symmetry group of the theory of special relativity. It was discovered that this is... 

Now in quantum theory the description of a physical system in the Heisenberg picture for a given observer O is by means of operators Q, which satisfy certain equations of motion and commutation rules with respect to O s frame of reference (coordinate system x). The above notion of an invariance principle can be stated alternatively as follows If, when we change this coordinate frame of reference (i.e., for observer O ) we are able to find a new set of operators that obeys the same equations of motion and the same commutation rules with respect to the new frame of reference (coordinate system x ) we then say that these observers are equivalent and the theory invariant under the transformation x - x. The observable consequences of theory in the new frame (for observer O ) will then clearly be the same as those in the old frame. 

The special theory of relativity builds on the idea that the laws of physics should be the same in all frames of reference that move relative to each other on straight trajectories with constant velocity, i.e., in unaccelerated motion. The second building block of the theory is the postulate that the velocity of light in the vacuum c is a fundamental constant and thus the same in all such inertial frames. The quantities defined in one inertial system are readily calculated in a different frame of reference by making use of a special prescription that states how time and space coordinates should change such that this postulate is satisfied the Lorentz transformation. 

---