Understanding rotational motion
Rigid bodies are often used to represent molecules with inactive vibrational degrees of freedom. In a rigid body, the locations of all constituent atoms are frozen with respect to the rigid-body coordinates. The intramolecular forces from bonds, angles, dihedrals, and impropers are assumed not to create any distortion of the rigid body. Therefore, for the purposes of MD, all intramolecular forces in a rigid body can be disregarded. Any force exerted on any constituent atom in a rigid body acts on the entire rigid body. In addition to changing its linear velocity, this force can create a torque on the rigid body and change its angular momentum and angular velocity. The total force on a rigid body 
composed of N particles is calculated using the following formula:
Using the displacement vector 
of particle i measured from the center of mass of the rigid body as the origin, the torque 
of the rigid body is calculated by the following formula:
The angular momentum 
of the rigid body is obtained from the time-integration of 
using the Velocity Verlet algorithm, as follows:
The angular velocity 
of the rigid body is obtained by a two-step procedure from 
using an intermediate step at a half-timestep 
and a second step at a full-timestep 
, which will be described in more detail when analyzing the dynamics of rigid bodies in Chapter 7, Understanding Fixes. The procedure is illustrated here:
Here, 
is the moment of inertia tensor of the rigid body.
For a system of point particles or rigid bodies, the average kinetic energy and the average speed or angular speed are determined by the system temperature. Therefore, controlling the temperature adequately is often essential when simulating a molecular system. The next section discusses the features of a molecular simulation determined by the temperature.
