Transformation Between Coordinate Systems

5 Key excerpts on "Transformation Between Coordinate Systems"

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  Modern Physics

  • Olga Moreira(Author)
  • 2019(Publication Date)
  • Arcler Press

    (Publisher)

  To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/. Modern Physics 2 in all practical situations it is a physical system which constitutes a reference frame. Can a quantum system be considered as a reference frame and, if so, which description would it give of the world? Here, we introduce a general method to quantise reference frame transformations, which generalises the usual reference frame transformation to a “superposition of coordinate transformations”. We describe states, measurement, and dynamical evolution in different quantum reference frames, without appealing to an external, absolute reference frame, and find that entanglement and superposition are frame-dependent features. The transformation also leads to a generalisation of the notion of covariance of dynamical physical laws, to an extension of the weak equivalence principle, and to the possibility of defining the rest frame of a quantum system. INTRODUCTION The state of a physical system has no absolute meaning, but is only defined relative to the observer’s reference frame in the laboratory. The same system may be associated to different states in different reference frames, which are normally related via some reference frame transformation. From a physical point of view, a frame of reference is an abstraction of an idealised physical system: for example, an ideal rigid body can serve as a reference frame to define relative spatial distances and orientations of other objects. In classical physics, a coordinate transformation is used to transform the description of the system under consideration between two different reference frames. These transformations include, for example, spatial rotations and translations in space and time or constant relative motion of the frames (e.g., Galilean tranformations). In general, the dynamical physical laws are invariant under some group of transformations.

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  A Modern Approach to Classical Mechanics

  • Harald Iro(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  Conceptually, a frame of reference comprises of a point of reference 1, together with devices for determining the position of some event with respect to the point of reference, and appropriately installed clocks for determining the time of occurrence of the event at any position 2. The mathematical counterpart of such a reference frame is a coordinate system (origin and axes) and a time scale, which together allow us to describe, for example, the motion of a particle at any time – its position r at time t, as well as its rate of change with time by and. In any particular reference frame, different coordinate systems may be chosen (e.g. Cartesian or spherical coordinates). When relating the observations of physical quantities in two different frames of reference, one has to state how to compare them: what do the views in the two systems have in common? A fundamental assumption of nonrelativistic, classical physics enters here: that the distance between two physically distinguished positions – e.g. between two bodies – is independent of the reference frame 3. However, changing the reference frame may also affect physical quantities, as in the example of particle collision the scattering angle or the velocity observed in the center of mass frame and in the laboratory frame. A change of coordinate system within a given frame, has, of course, no influence on the physical processes; only the (mathematical) description is affected by the choice 4. The coordinate system is often considered to be the reference frame, but strictly speaking one should distinguish between these two notions. Of course, a change of reference frame is accompanied by a coordinate transformation (but not, in general, vice versa)

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  Fundamentals of Modern Mathematics

  A Practical Review

  • David B. MacNeil(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  Chapter 7 TRANSFORMATIONS AND TRANSFORMS

  The words transformation and transform, to mention them in the order in which they are treated in this chapter, have a number of uses in mathematics. In fact, the word transformation is essentially synonymous with function, operator, correspondence, etc. However, partly for historical reasons and partly for definiteness, the different words are used in different settings, which reduces somewhat the use of the word transformation.

  One of its general uses is in connection with changes in coordinate systems. Such changes may be made in the same system, as by translation or rotation of the axes of a rectangular coordinate system; or they may involve a change in the coordinate system itself, e.g., change from rectangular coordinates to spherical coordinates, cylindrical coordinates, or to any of the numerous other kinds of coordinate systems.

  Such changes in system naturally change an equation which is expressed in the system. In fact, such changes in equations are one of the reasons why changes in systems are often made, for as is shown in this chapter, an equation can often be simplified by such changes, and therefore be made more useful in computation, and more readily adapted to other mathematical operations.

  The word transform also has a number of specific present-day uses. Its application to operations with matrices can be shown to follow from the methods of Chapter 2 . For if A, B, and X are three matrices then B = X−1 AX is the transform of A by X (X being non-singular).

  There is also an application of the word transform to integral equations since in the equation

  f(y) is called a transform of F(x

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  Kinematic Geometry of Surface Machining

  • Stephen P. Radzevich(Author)
  • 2007(Publication Date)
  • CRC Press

    (Publisher)

  Equations similar to that above are valid for other operators Tr ( a y , Y ) and Tr ( a z , Z ) of the coordinate system transformation (Figure 3.3b and Figure 3.3c). Suppose that a point P on a rigid body goes through a translation describ-ing a straight path from P 1 to P 2 with a change of coordinates of ( a x , a y , a z ). This motion can be described with a resultant translation operator Tr ( a , A ): Tr A ( , ) a a a a x y z =             1 0 0 0 1 0 0 0 1 0 0 0 1 (3.5) The operator Tr ( a , A ) can be interpreted as the operator of translation along an arbitrary axis A . An analytical description of translation of the coordinate system X 1 Y 1 Z 1 along an arbitrary axis A to the position of X 2 Y 2 Z 2 can be composed from Figure 3.4. a 2 (b) (c) a x a z Y 1 Y 1 Y 2 Y 2 Y 1 Y 2 Z 1 Z 1 Z 2 Z 2 Z 1 Z 2 X 1 X 2 X 2 X 1 X 1 X 2 (a) FIGURE 3.3 Analytical description of the operators of translations Tr ( a x , X ), Tr ( a y , Y ), Tr ( a z , Z ) along the coordinate axis. Applied Coordinate Systems and Linear Transformations 69 The operator Tr ( a , A ) of translation of that kind can be expressed in terms of the operators Tr ( a x , X ), Tr ( a y , Y ), and Tr ( a z , Z ) of elementary translations Tr A Tr Tr Tr ( , ) ( , ) ( , ) ( , ) a a Z a Y a X z y x = ⋅ ⋅ (3.6) Evidently, the axis A is always the axis through the origin. Any coordinate system transformation that does not change the orienta-tion of a geometrical object is an orientation-preserving transformation , or a direct transformation. Therefore, transformation of translation is an example of a direct transformation. 3.2.3 Rotation about a Coordinate Axis Rotation of a coordinate system about a coordinate axis is another major lin-ear transformation used in the theory of surface generation. Rotation of the coordinate system X 2 Y 2 Z 2 about the axis of the coordinate system X 1 Y 1 Z 1 is illustrated in Figure 3.5.

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  Introduction to General Relativity

  International Series of Monographs in Natural Philosophy

  • H. A. Atwater(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  The coordinate t It will be recognized that here we treat time as a parameter, upon which the space coordinates depend, rather than as an independent variable of a four-dimensional continuum. Expressions in which the coordinates appear as functions of time may, in fact, be regarded as descriptions of lower-dimensional subsurfaces in the four-dimensional space. INTRODUCTION 5 transformation of eqns. (1.2) may be described by the matrix of transfor-mation coefficients: f 1 000 a cm — V 1 0 0 óx 0010 0001 (1.2a) The Galilean transformation of eqns. (1.2) includes the expression of the Newtonian assumption that the time scale is universal, or that time flows equally for all observers. For a mass point m in coordinate frame É which is constrained about the point r 0 = (x0 , Yo, by a spring force with spring constant k the equations of motion [eqns. (1.1)] take the form mx = —k(x — x o ), my = — k(Ý — 5k), (1.3) mz = —k(z — When the Galilean transformation, eqns. (1.2), is applied to these equa-tions of motion using x = (x — Vt), x o = (c o — N ~ , y0 = Yo, Zo = z o etc., we find mx = —k(x — x o ) , n 1 ~~ = —k(y — yo), (1.4) m2 = —k(z — z u ). The identity of the character of eqns. (1.4) with eqns. (1.3) is an expres-sion of the covariance of Newton's laws under Galilean transformation. To the physicists of the nineteenth century the Galilean transformation (1.2) was accepted as being manifestly correct, since it seemed to be an explicit and indisputable statement of what was actually happening, during the relative translation of two coordinate frames. A conflict with experiment appeared, however, when this transformation was applied to problems involving the propagation of light. t It may be observed that in calculating the matrix element bt M/ c 1 we associate the numerator of the differential quotient with the rows of the matrix, and the deno-minator with its columns.

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