||Message]
Category:
A possible way of specifying the orientation of a rigid body is to give orientation of body fixed axes w.r.t. a space fixed axes. Euler angles are a useful set generalized coordinates to specify orientation of the body axes relative to a space fixed axis.
A rigid body is defined as a collection of particles such that the distances between them is fixed. In order to use the Lagrangian formalism to describe the motion of a rigid body, we must find the number of generalized coordinates required to specify its configuration. That a rigid body has six degrees of freedom can be seen as follows.
If we take an arbitrary point on the rigid body, we need three coordinates to specify is location. Once a point of the body is held fixed, any displacement of the the body, by Euler' theorem,is a rotation about some line. Thus specifying the axis of rotation and angle of rotation fixes the configuration of the rigid body. A unit vector specifies a line in space and the number of required independent parameters is two. The rotation about the axis requires one more parameter, {\it viz. } the angle of rotation. Thus in all we need six parameters to specify configuration of a rigid body.
Out of these six degrees of freedom three correspond to translations and three correspond to rotational motion.
In order to setup the Lagrangian for motion of rigid body we need to choose six generalized coordinates. By specifying a chosen coordinate system, fixed in the body, completely specifies the body configuration. This requires three coordinates for specifying the origin of the body axes and three parameters to specify the orientation of the body fixed axes.
Thus we are led to two sets of coordinate systems, {\tt space fixed axes} and {\tt body fixed axes}. The body fixed axes are completely specified by giving the coordinates of the origin and the orientation w.r.t the space fixed axes.
One way to parameterize an orientation of the body fixed axes is to give the axis and angle of rotation. The axis is specified by a unit vector \(\hat{n}=(n_1, n_2,n_3)\). This specification of rotations, by unit vector and an angle, is not convenient to be used as generalized coordinates. This is because the components \(n_1, n_2, n_3\) are not independent.
Several generalized coordinates to specify orientation of the body axes relative to a space fixed axis with common origin are available, Euler angles being a useful set to describe the motion of a rigid body.
