Classical Mechanics - Rigid Body Mechanics : Euler’s Equations - Lagrange’ s Method

An alternative approach to Euler’s equations is the Lagrange’s method using the Euler’s angles as the generalized coordinates. When one point of the body is fixed,only the rotational motion is possible and the Euler’s angles completely describe the orientation of the rigid body. In such a situation, the Lagrangian L,is a function of three Euler’s angles only. Thus the kinetic energy will only involve these coordinates and likewise potential energy also. It will be remembered that the generalized forces corresponding to these angular coordinates will be the components of the applied torque along the axis of rotation. Taking these facts under account,Lagrange’s equations are set up and Euler’s equations are obtained.