Let's consider I have a particle moving on the x,y plane. On this particle acts the Lorentz force, however it does not necessary perform a rotation by the axes origin ( x , y ) = ( 0 , 0 ) . I would like to re-write the Lorentz force in the "global" cylindrical coordinate system where the origin ( x , y ) = ( 0 , 0 ) is the same as the r=0. Of course the particle motion will be described by R = r , ϕ , z and u = u r , u ϕ , u z in that coordinate system. Should I add an inertial force in this system, such as m d u d t = F L + F i n t , in my description?

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Let&#039;s consider I have a particle moving on the x,y  plane. On this particle acts the Lorentz force, however it does not necessary perform a rotation by the axes origin   (  x  ,  y  )  =  (  0  ,  0  ) . I would like to re-write the Lorentz force in the &quot;global&quot; cylindrical coordinate system where the origin   (  x  ,  y  )  =  (  0  ,  0  ) is the same as the r=0. Of course the particle motion will be described by   R  =  r  ,  ϕ  ,  z and   u  =      u    r    ,      u          ϕ        ,      u    z   in that coordinate system. Should I add an inertial force in this system, such as   m            d      u              d      t        =      F    L    +      F          i      n      t      , in my description?

Gael Irwin · Accepted Answer

First you write the equations of motion in Inertial system             R      →        =  [  x  ,  y      ]    T                      (1)                    m                                                                    R                ¨                                      →                          =                  F          L                (                              R            →                          ,                                                                              R                  ˙                                            ,              t              ,                                                q                  ¯                                                      →                          )                  F    L   is the Lorentz force and             q      ¯       the electric chargenow you can transformed equation (1) to polar coordinates with:            R      →        (  x  ,  y  )  ↦            R      →        (  r  ,  ϕ  )                              R          ˙                    →        =            ∂                        R          →                            ∂                        q          →                                                  q          ˙                    →                                    R          ¨                    →        =            ∂                        R          →                            ∂                        q          →                                                  q          ¨                    →        +                    f        z            →        (            q      →        ,                              q          ˙                    →        )with:             q      →        =  [  r  ,  ϕ      ]    T     equation (1)                    (2)                    m                          (                                    ∂                                                R                  →                                                                    ∂                                                q                  →                                                                                                                          q                  ¨                                            →                                +                                                    f                z                            →                                (                                    q              →                                ,                                                                      q                  ˙                                            →                                )          )                =                  F          L                (                              q            →                          ,                                                            q                ˙                                      →                          ,        t        ,                              q            ¯                          )            multiply equation (2) from the left with       J    T    =            (                        ∂                                    R              →                                                ∂                                    q              →                                          )        T   we obtain :                    (3)                    m                          J          T                        J                                                                    q                ¨                                      →                          =                  J          T                                  F          l                −        m                          J          T                                                            f              →                                z                    Equation (3) are the equations of motion in polar coordiantes

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