Change of Frame

Change of frame from the original frame to a new starred one is given by 

Cartesian components of the same event in a Cartesian coordinate system fixed with an old (original) or new (starred) frame, respectively in the Cartesian system of a new frame at given instant t, and Q x are the positions of origin and of event (seen in the Cartesian system of the old frame), respectively, cf. Fig. 3.1. 

Transformations (3.25) and (3.26) follow from the expected properties of the change of frame in classical physics the distance between two simultaneous events 

It is evident from (3.25) and (3.26) that b is the time shift in the origin of the time axes, c is the shift in the origins of the Cartesian systems and Q (from full orthogonal group, cf. Rem. 8) expresses the rotation (detQ = 1) or reflection (detQ = —1) of the starred frame relative to the original one. We also note the inversions of change 

The special case of (3.25) is the Galileo transformation where function c(t) is linear and Q is a constant, i.e., frames are moving each to the other with constant velocity and differ by a constant angle (or also by inversion). A special set of frames must be noted—inertial frames which contain the frame formed by distant stars and those obtained from it by Galileo transformation (cf. Sects. 3.3, 4.3 in many applications the frame fixed with earth surface may be taken as an approximately inertial one). Their typical property is zero inertial acceleration (3.48). 

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Relations (b), (c) inspire in Sect. 3.2 the more general notion of changes of frame and frame... 

Use of a Ml or proper orthogonal group puts the additional property of preservation of right- or left handedness on the change of frame some authors [12, 23-26] (motivated usually... 

A more special case with c = o and Q constant is physically trivial because it expresses the change of coordinate system only. Therefore, a change of coordinates (in Rem. 4) is not the same as the much more general change of frame (where time and its transformation (3.26) and shifts in origins are moreover considered). 

Many quantities used in the following considerations are called objective ot frame-indifferent, if they are invariant in the change of frame (3.25), (3.26) as follows (because this change contains rotations and/or inversions of corresponding Cartesian systems as a very special case (cf. Fig. 3.1), the following definition is motivated by (b), (c) of Rem. 4) ... 

Objective or frame-indifferent scalar a, vector a and (second order) tensor A transform by the change of frame on scalar a, vector a and tensor A as follows ... 

The reference configuration and its properties (like particles and bodies in reference) are not infiuenced by changing of frame (this affects actual refo ences only). 

For the other defined quantities, we decide from their definitions assuming that the definition itself is not influenced by change of frame (i.e., definitions are the same in any frame). Not only those, but in fact all relations between quantities (e.g., those from Sect. 3.1) are valid also for new (starred) frame, i.e., for new starred quantities if we use t, x (3.25), (3.26) simultaneously. This is evident from the fact that the frame used for actual reference (say in Sect. 3.1) was chosen quite arbitrarily. Cf. also end of this section. 

Applying the change of frame (3.25), (3.26) to the above definitions of Sect. 3.1 and, using these precepts, we can decide about objectivity or nonobjectivity (frame indifference or not) of the following quantities (more detailed proofs of some of them are written in the footnote-sized script below) the remainder from the next sections may be proved analogously. 

As we noted in the precepts above the remaining relations of Sect. 3.1 are also valid in all frames, e.g., (3.21), (3.22) or Reynolds theorem (3.24). Because of no influence of the change of frame on the reference configuration (and material points), no such influence may be also expected on material volume V, material surface 9V (they behave as objective scalars), and the outside normal n should be an objective vector. 

But this is not all. Using again the change of frame with arbitrary Q in the constitutive equations (3.121) we have (note that here all dependent and independent variables are objective we can regard F and GradF as objective vectors, cf. (3.49), (3.51))... 

Cf. (3.18), Rem. 5 and deduction of (3.25). Because the change of frame describes the change of frame in a rigid motion to another one the result (3.223) is intuitively clear. Formally, inserting rigid motion from Rem. 5 into (3.25) we seek the (starred) frame in which x = X (and therefore V = o, i.e., (3.223)) through the body. It may be seen that this need the change of frame by time functions Q = and c = — y. 

A fundamental principle of classical physics is that material properties must be independent of the frame of reference or observer. This axiom is commonly called the principle of material frame-indifference or objectivity [270, pp.41-44]. This principle states that constitutive equations (discussed in greater detail in Section 4.2.3) must be invariant under changes of frame of reference. Under the motion defined by... 

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