Motion in a Central Field: Conservation of Angular Momentum and Confined Motion in a Plane

Introduction
In classical mechanics, a central field refers to a force that is directed towards or away from a fixed point, known as the center. Examples of central fields include gravitational and electrostatic forces. When a particle moves in a central field, certain quantities are conserved, while others may vary. In this case, we will explore the conservation of angular momentum and the confinement of motion in a plane.

Conservation of Angular Momentum
Angular momentum is defined as the cross product of the particle's position vector and its momentum vector. In a central field, the force acting on the particle is always directed towards or away from the center, which means that it is always perpendicular to the position vector. As a result, the torque acting on the particle is zero, and hence the angular momentum is conserved.

This conservation of angular momentum can be mathematically expressed as L = r x p = constant, where L is the angular momentum, r is the position vector, and p is the momentum vector. Since the direction of the position vector and momentum vector will change as the particle moves, the magnitude of the angular momentum will remain constant.

Confined Motion in a Plane
Since the force in a central field is always directed towards or away from the center, the motion of the particle is confined to a plane. This can be understood by considering the motion in terms of polar coordinates. The radial component of the motion determines the distance from the center, while the angular component determines the direction of the motion within the plane.

As the particle moves, the direction of the force and hence the direction of the acceleration will change. However, since the force is always perpendicular to the velocity vector, the motion will remain confined to a plane. This can be visualized as the particle moving along a trajectory that lies in a single plane.

Conclusion
In summary, when a particle moves in a central field, the conservation of angular momentum and the confinement of motion in a plane are observed. The angular momentum is conserved due to the perpendicularity of the force and the position vector, resulting in a constant magnitude of angular momentum. Additionally, the motion of the particle is confined to a plane due to the perpendicularity of the force and velocity vectors. These characteristics are fundamental to understanding the behavior of particles in central fields and have important implications in various areas of physics.