Exercise:
Newton's second law in the form vec F fractextrmdvec ptextrmdt dotvec p is still correct in relativistic physics. abcliste abc Show that when the force acting on a particle is parallel to its velocity it can be written as F gamma^ m dot v gamma^ m a abc Show that the general relation between the force vector and the accelerations parallel and perpicular to the motion of the particle is vec F gamma^ m vec a_parallel + gamma m vec a_perp abcliste

Solution:
abcliste abc The relativistic momentum is p gamma m v The derivative with respect to time is dot p mleftdotgamma v+gammadot vright The derivative of the Lorentz factor with respect to time is fractextrmdgammatextrmdt fractextrmdgammatextrmdv fractextrmdvtextrmdt fractextrmdtextrmdvleft-leftfracvcright^right^-/ dot v -fra vleft-leftfracvcright^right^-/left-fracvc^right fracv dot vc^ gamma^ Putting everything together we find dot p gamma m dot v leftgamma^fracv^c^+right gamma m dot v leftfracfracv^c^-fracv^c^+right gamma m dot v leftfracv^c^-v^+right gamma m dot v left fracv^+c^-v^c^-v^right gamma m dot v left frac-fracv^c^right gamma m dot v gamma^ gamma^ m dot v gamma^ m a abc The derivative of the momentum vector with respect to time is dot vec p mleftfractextrmdgammatextrmdt vec v + gamma fractextrmdvec vtextrmdtright The derivative of the Lorentz factor can be derived in a similar way as before but we have to take o account that v^ v_x^+v_y^+v_z^ so according to the chain rule we have the following expression: dot gamma fractextrmdgammatextrmdv_x dot v_x + fractextrmdgammatextrmdv_y dot v_y + fractextrmdgammatextrmdv_z dot v_z With fractextrmdgammatextrmdv_x fractextrmdtextrmdv_xleft-fracv_x^+v_y^+v_z^c^right^-/ frac-left-fracv_x^+v_y^+v_z^c^right^-/ leftfrac- v_xc^right fracv_xc^ gamma^ and the anologous expressions for the derivative with respect to v_y and v_z we find for dot gamma dot gamma gamma^ fracv_x dot v_x+v_y dot v_y + v_z dot v_zc^ fracgamma^c^ vec v dotvec v It follows for the derivative of the momentum vector dotvec p m leftfracgamma^c^vec v dotvec vvec v + gamma dotvec v right gamma m leftfracgamma^c^ vec v vec a vec v+vec aright We can express the acceleration as the e of the accelerations parallel and perpicular to the motion: vec a vec a_parallel + vec a_perp Since vec v perp vec a_perp we have vec v vec a vec v vec a_parallel + vec a_perp vec v vec a_parallel and since vec v and a_parallel are parallel the vector vec v vec a_parallel vec v has the same direction as a_parallel and magnitude v^ a_parallel so it can be written as vec v vec a_parallel vec v v^ vec a_parallel The force vector is thus vec F dotvec p gamma m leftgamma^fracv^c^ vec a_parallel + a_parallel + a_perpright gamma m leftgamma^ fracv^c^+right vec a_parallel + gamma m vec a_perp gamma^ m vec a_parallel + gamma m vec a_perp The result shows that the ratio between force and mass is not the same for forces acting parallel and perpicular to the motion of the particle. abcliste