Central force

A central force is a force applied to a fixed point ( the center of force ) is always related, ie to Z Z to or from points away.

Many central forces ( conservative ) gradient to a spherically symmetric central potential (also central field, see below). In this article, however, also non-conservative central forces are treated, the particular must not have radial symmetry.

Gravity and the Coulomb force are examples of conservative central forces. Strictly speaking, it depends on the reference frame, if the above definition is correct; it is about the gravity only in the center of mass system ( and all relative to these quiescent systems ) is a central force.

Conservation of angular momentum

Under the influence of a general central force, the angular momentum remains: get a mass point in the reference system with the origin. For the angular momentum

Applies namely

Which is just used in the last step that the force

Is parallel to the position vector.

This is just the content of Kepler's second law, for which is required only as a condition that the force is in the radial direction.

From the conservation of angular momentum follows that the movement is in the plane in which the initial values ​​are of and.

Central potential

At a central potential is meant a potential which depends only on the distance from the center of force. It is therefore necessary. From a central potential only central force fields can be derived which have no angular dependence, which are thus spherically symmetric.

This becomes clear when considering the operator in spherical coordinates look at

Thus, a force field only shows in the radial direction, and must be. But if not dependent on the angles, it is not.

General central forces

One consequence of this is that angle-dependent central force fields are not conservative; there is no central potential, from which they can be derived. In them, the work done depends on the path. It is then true law of areas ( conservation of angular momentum ), but not the energy conservation.

Demarcation of the centripetal force

The power center is for ellipse, parabola and hyperbolic in one of the foci of the web and is to be distinguished from the center of the (local) path curvature. Both points are only correct for circular orbits coincide with each other, then the central force coincides with the centripetal force for the train. For the said general paths directed to the focal point central force is split into a normal component to the center of the (local) curvature circle and a tangential component in the web direction. The latter component ensures, for example, that a planet moves faster at perihelion than at aphelion.

Central movement

The path of a particle at a central field is valid classical mechanics in one plane. Key systems, which are modeled with a central motion are:

• The atom with its electrons: The behavior of the electrons is explained by the solution of a quantum-mechanical central problem.
• Binary Stars: A binary star system is an example of a two-body problem. This is interpreted as the motion of two bodies around their common center of gravity. Depending on the required accuracy is, for example, classical mechanics and general relativity are used.
• Approximately the solar system: proximity, the motion of the planets in the solar system can be considered as motion in the gravitational field of the sun. However, the body in the solar system have self- gravitational fields and thus interfere with the movement of another body, so that a planetary orbit can not be explained just by the motion in the gravitational field of the sun.