# Central force

A central force is a force that is always related to a fixed point (the center of force ), i.e. points towards or away. ${\ displaystyle Z}$ ${\ displaystyle Z}$ ${\ displaystyle Z}$ Many central forces are (conservative) gradient fields to a spherically symmetrical central potential (also central field , see below). However, this article also deals with non-conservative central forces, which in particular do not have to have radial symmetry.

The gravitation and the Coulomb force are examples of conservative central forces. Strictly speaking, it depends on the frame of reference whether the above definition applies; for example, gravity is only a central force in the center of gravity system (and all relative to these systems at rest).

## Angular momentum conservation

Under the influence of a general central force, the angular momentum of a mass point is retained in the reference system with the origin . For the angular momentum ${\ displaystyle {\ vec {L}}}$ ${\ displaystyle Z}$ ${\ displaystyle {\ vec {L}}: = {\ vec {r}} \ times {\ vec {p}} = {\ vec {r}} \ times m \; {\ dot {\ vec {r} }}}$ namely applies

${\ displaystyle {\ frac {\ mathrm {d} {\ vec {L}}} {\ mathrm {d} t}} = m \ left ({\ dot {\ vec {r}}} \ times {\ dot {\ vec {r}}} + {\ vec {r}} \ times {\ ddot {\ vec {r}}} \ right) = {\ vec {r}} \ times {\ vec {F}} = {\ vec {0}}}$ ,

where the last step uses that force

${\ displaystyle {\ vec {F}} ({\ vec {r}}) = F (r) {\ frac {\ vec {r}} {r}} = F (r) {\ vec {e}} _ {r}}$ is parallel to the position vector.

That is precisely the content of Kepler's second law , for which the only requirement is that the force points in the radial direction.

From the conservation of angular momentum it follows that the movement remains in the plane in which the initial values ​​of and lie. ${\ displaystyle {\ vec {r}}}$ ${\ displaystyle {\ dot {\ vec {r}}}}$ ## Central potential

A central potential is understood to be a potential that only depends on the distance to the center of force. So it applies . From a central potential, only central force fields can be derived that have no angular dependency, i.e. that are spherically symmetrical. ${\ displaystyle r}$ ${\ displaystyle V ({\ vec {r}}) = V (| {\ vec {r}} |) = V (r)}$ This becomes clear when you look at the Nabla operator in spherical coordinates ${\ displaystyle {\ vec {\ nabla}}}$ ${\ displaystyle {\ vec {\ nabla}} = {\ vec {e}} _ {r} {\ frac {\ partial} {\ partial r}} + {\ vec {e}} _ {\ theta} { \ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} + {\ vec {e}} _ {\ varphi} {\ frac {1} {r \ sin \ theta}} {\ frac {\ partial} {\ partial \ varphi}}}$ .

So that a force field only points in radial direction, must and be. But if it doesn't depend on the angles, then it wo n't. 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. The law of areas (conservation of angular momentum) then applies, but not the conservation of energy. ${\ displaystyle {\ vec {F}} = - q {\ vec {\ nabla}} \ Phi}$ ${\ displaystyle {\ frac {\ partial} {\ partial \ theta}} \ Phi = 0}$ ${\ displaystyle {\ frac {\ partial} {\ partial \ varphi}} \ Phi = 0}$ ${\ displaystyle \ Phi}$ ${\ displaystyle {\ vec {F}}}$ ## Differentiation from centripetal force

The centripetal force is determined from the speed and the curvature of the movement of a body at its current location and points to the center of the circle of curvature, which does not have to coincide with the physical center of force. The center of force for elliptical, parabolic and hyperbolic orbits lies in one of the focal points of the orbit. Both points coincide only for circular paths for which the central force then coincides with the centripetal force of the path. For the general paths mentioned, the central force directed towards the focal point is to be divided into a normal component to the center of the (local) circle of curvature and a tangential component in the direction of the path. The latter component ensures z. B. for a planet slowing down on the way from perihelion to aphelion .

## Central movement

The path of a mass point in a central field lies in one plane if classical mechanics is valid . Important systems that are modeled with a central movement are:

• the atom with its electrons : The behavior of the electrons is explained by the solution of a quantum mechanical central problem (" hydrogen problem ").
• Binary Stars : A binary star system is an example of a two-body problem . This is understood as the movement of two bodies around their common center of gravity . Depending on the required accuracy, classical mechanics or general relativity , for example, are used.
• approximately the solar system : The movement of the planets in the solar system can approximately be regarded as movement in the gravitational field of the sun . However, the bodies in the solar system themselves have gravitational fields and thus disturb the movement of the other bodies, so that a planetary orbit cannot be exactly explained by the movement in the gravitational field of the sun.