Galilei Covariance of Newtons Laws

Covariance of equations shall denote their invariance in form under coordinate transformations. Covariance of Newton s first law, Xi = 0 (i = 1,2,3), under general Galilean transformations given by Eqs. (2.14) and (2.15) is obvious, since 

Newton s first law is therefore valid in one specific IS if and only if it is valid in all other inertial frames related to IS via Galilean transformations. Newton s second law in IS is in general given by 

Equations (2.19) and (2.20) have exactly the same form, i.e., Newton s equation of motion is indeed covariant under Galilean transformations. These two equations describe the same physical situation with respect to two different inertial frames of reference. Although the physical vectorial force is of course the same in both frames of reference, F = F, its components F, and F- are in general different functions of their arguments. This relationship is given by the second equality of Eq. (2.20). 

4 Scalars, Vectors, and Tensors in Three-Dimensional Space 

Newtonian mechanics is set up in flat and Euclidean three-dimensional space R equipped with a linear time axis. The most general symmetry transforma- 

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