In classical mechanics, the state of a system is described by specifying both the position and velocity of an object. This is known as a phase space representation, where each point in the phase space corresponds to a unique state of the system.
The reason for using two state vectors, one for position and one for velocity, is rooted in the fundamental principles of classical mechanics. These principles include Newton's laws of motion, which describe how the motion of an object is influenced by forces acting upon it.
The position vector provides information about the location of the object in space at a particular instant in time. It gives us an understanding of where the object is located relative to a chosen reference point.
On the other hand, the velocity vector gives information about how the position of the object changes with respect to time. It represents the rate at which the object's position is changing. By knowing both the position and velocity vectors, we can determine the instantaneous state of the object and predict its future motion.
The use of position and velocity vectors allows us to accurately describe and analyze the behavior of objects in classical mechanics. It provides a comprehensive representation of the state of a system, enabling us to study and understand various physical phenomena, such as the motion of particles, the behavior of celestial bodies, and the interactions between objects.