Central force

In, a central force on an object is a that is directed along the line joining the object and the origin:
 * $$ \vec{F} = \mathbf{F}(\mathbf{r}) = F( \mathbf{r} ) \hat{\mathbf{r}} $$

where $$ \scriptstyle \vec{ \text{ F } } $$ is the force, F is a, F is a scalar valued force function, r is the , ||r|| is its length, and $$ \scriptstyle \hat{\mathbf{r}}$$ = r/||r|| is the corresponding.

Not all central force fields are nor. However, it can be shown that a central force is conservative if and only if it is spherically symmetric.

Properties
Central forces that are conservative can always be expressed as the negative of a :-
 * $$ \mathbf{F}(\mathbf{r}) = - \mathbf{\nabla} V(\mathbf{r})\text{, where }V(\mathbf{r}) = \int_{|\mathbf{r}|}^{+\infin} F(r)\,\mathrm{d}r$$

(the upper bound of integration is arbitrary, as the potential is defined an additive constant).

In a conservative field, the total ( and potential) is conserved:
 * $$E = \frac{1}{2} m |\mathbf{\dot{r}}|^2 + V(\mathbf{r}) = \text{constant}$$

(where ṙ denotes the of r with respect to time, that is the ), and in a central force field, so is the :
 * $$\mathbf{L} = \mathbf{r} \times m\mathbf{\dot{r}} = \text{constant}$$

because the exerted by the force is zero. As a consequence, the body moves on the plane perpendicular to the angular momentum vector and containing the origin, and obeys. (If the angular momentum is zero, the body moves along the line joining it with the origin.)

It can also be shown that an object that moves under the influence of any central force obeys Kepler's second law. However, the first and third laws depend on the inverse-square nature of and do not hold in general for other central forces.

As a consequence of being conservative, these specific central force fields are irrotational, that is, its is zero, except at the origin:
 * $$ \nabla\times\mathbf{F} (\mathbf{r}) = \mathbf{0}\text{.}$$

Examples
Gravitational force and are two familiar examples with $$ F( \mathbf{r} ) $$ being  only. An object in such a force field with negative $$ F( \mathbf{r} ) $$ (corresponding to an attractive force) obeys.

The force field of a spatial is central with $$ F( \mathbf{r} ) $$ proportional to r only and negative.

By, these two, $$ F( \mathbf{r} ) = -k/r^2 $$and $$ F( \mathbf{r} ) = -kr $$, are the only possible central force fields where all bounded orbits are stable closed orbits. However, there exist other force fields, which have some closed orbits.