Motion in moving systems

                                                   Renato Iraldi

                                      Escuela de física y matemáticas

                                     Universidad central de Venezuela

                                               Caracas, Venezuela.   

                                               I Introduction

            Newtonian physics is described considering absolute space and time. Vectors in this space are defined in two different althought equivalent forms: the geometrical method and the algebraic method. These methods are not totally equivalent since in the algebraic method a reference frame with a coordinate system is needed. A vector, in this method, is a set of three number asociated to the coordinate system of the reference frame and subject to a definite transformation law when a change of  coordinates is made. The geometrical method does not need a coordinate system. Rigorously speaking the representations of vectors are made by means of different mathematicals objects; the abstract geometrical vectors, and, a set of three numbers referred to a particular coordinate system of a reference frame .

            The simultaneus use of both definitions may produce dificulties for the non advised students. An example of this is seen, in nearly all books of mechanics, when treating the motion in moving coordinate systems. Particularly, it is a common feature to find that there is a relation between temporal derivation in systems that has a relative rotation. This relation is written:

d/dt  = d^*/dt    + w´

meaning that the operation of taking the time derivative of any vector referred to the non- rotating system is equivalent to operate with de right hand side of this equation in the rotating system. Generally, this relation is shown by a geometrical hand waving argument that may be confusing.

            It may be usefull to clarify these difficulties giving a rigorous mathematical treatment of the problem of moving reference frames without any geometrical arguments. The use of  Dirac’s notation, althought not indispensable, will simplify the treatment. No previuos knowledge of this notation in required.

                                   II Dirac’s notation

            An abstract vector a, is represented in the algebraic method by three numbers a_1, a_2, a₃;  referred to three unitary vectors  e₁= êe₁>, e₂ = êe₂>, e₃ = çe₃>;  forming a basis or coordinate system íe_ný.  a₁ = e₁ · a, a₂ = e₂·a  y a₃ = e₃·a.  In Dirac notation a₁ = <e₁ça>, a₂ = <e₂ça>  and a₃ = <e₃ça> these numbers are called the coordinates of the vector ça>. Here the scalar product of two vectors  a · b is denoted following Dirac’s notation <aêb>.

            Since the vectors form a basis we have

S₁³ çe_n><e_nê= I,  unity operador                        ...(1)

this operator applied to a vector take the three projection along the unitary basis vectors and multiply it by the unitary vectors. By means of this operator we can write

  ça>  = S₁³ çe_n><e_nça> = êe₁><e₁ça> + êe₂><e₂êa> + çe₃><e₃êa>.

or in a different reference system

ça> = êe₁><e₁ça> + êe₂><e₂êa> + çe₃><e₃êa>.

            The coordinates belonging to two different frames are related by a 3x3  transformation matrix of which <e_nêe_i> are the elements.  This matrix is the rotation matrix formed by the orthogonal projections of the unitary basis vectors of one frame on the unitary basis vector of the other frame .

                                          <e_nêa> = S_i=1³ < e_nêe_i><e_iïa>   ...(2)

            The algebraic representation of a vector is formed by  the coordinates  <e_nêa>, and a basis íe_ný of a coordinates system.

                                êe_n> d/dt <e_nçr> =  êe_n> d/dt <e_nêR> +  êe_n> d/dt <e_nêr’>

or

                                 êe_n><e_nçv> =  êe_n> <e_nêU> +  êe_n> d/dt <e_nêr’>

êe_n> d/dt <e_nêr’>, is the velocity of the particule relative to the origin O’ of  S’ expressed in the basis   íe_ný of  S. If we are interested in the relation between the coordinates of the velocity of the particule as measured by an observer in S and an observer in S’,  we introduce the unity operator êe_i><e_iç to obtain

                           êe_n><e_nçv> =  êe_n> <e_nêU> +  êe_n> d/dt <e_nêe_i><e_i êr’>

As the orientation of the basis  S’ and S are fixed then:  d/dt <e_nêe_i> = 0, and

                                 êe_n><e_nçv> = êe_n> <e_nêU> +   êe_n>  <e_nêe_i> d/dt <e_i êr’>     

 Where   d/dt <e_i êr’> = <e_i êv’> is the derivative of the components of r’ referred to the basis íçe_j>ý, then:

                                 êe_n><e_nçv> = êe_n> <e_nêU> +  êe_n>  <e_nêe_i>  <e_i êv’>   

This is the same as eq.(4) in the algebraic form.

Diferentiation of eq.(4) give similarly

                                          êa> = êA> +  êa’>

or in algebraic form

                    êe_n><e_nça> = êe_n> <e_nêA> +  êe_n>  <e_nêe_i> <e_i êa’>                                 

            êa> is the acceleration of the particule relative to O, and is the measured acceleration for an observer in S,  êA> is the acceleration of O’ relative to O, physicaly is the acceleration of the observer in S’ as observed by the observer in S; and   êa’> is the acceleration of the particule relative to O’, or the measured acceleration for an observer in S’.

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