Degrees of freedom — UPSC Physics Optional CSE 2009 Paper I

Show that for any rigid body consisting of at least three particles, not arranged in one straight line, number of independent degrees of freedom is six. Define Euler's angles \theta, \phi and \psi to describe the configuration of such a rigid body. Consider two frames of reference, one fixed to the body and the other to the space defined as S' = (x', y', z') and S = (x, y, z) respectively. Show that the angular momentum (\vec{L}) of the rigid body in the two frames are related by \left( \frac{d\vec{L}}{dt} \right)_S = \left( \frac{d\vec{L}}{dt} \right)_{S'} + \vec{\omega} \times \vec{L} where \vec{\omega} is the angular velocity of rotation.