Central force

In classical mechanics, a **central force** on an object is a force that is directed along the line joining the object and the origin:

F
→

=

F

(

r

# )
F
(

r

)

r

^

{\displaystyle \vec{F} = \mathbf{F}(\mathbf{r}) = F( \mathbf{r} ) \hat{\mathbf{r}} }

where

 F
→

{\displaystyle \scriptstyle \vec{ \text{ F } } }

r

^

{\displaystyle \scriptstyle \hat{\mathbf{r}}}

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 gradient of a potential energy:

F

(

r

# )
−

∇

V
(

r

)
, where
V
(

r

# )

∫

r

∞

F
(
r
)

d

r

{\displaystyle \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 up to an additive constant).

In a conservative field, the total mechanical energy (kinetic and potential) is conserved:

# E

1
2

m

r
˙

2

V
(

r

# )
constant

{\displaystyle E = \frac{1}{2} m |\mathbf{\dot{r}}|^2 + V(\mathbf{r}) = \text{constant}}

(where **ṙ** denotes the derivative of **r** with respect to time, that is the velocity), and in a central force field, so is the angular momentum:

L

=

r

×
m

r
˙

=
constant

{\displaystyle \mathbf{L} = \mathbf{r} \times m\mathbf{\dot{r}} = \text{constant}}

because the torque 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 Kepler's second law. (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 Newton's law of universal gravitation 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 curl is zero:

∇
×

F

(

r

# )

0

.

{\displaystyle \nabla\times\mathbf{F} (\mathbf{r}) = \mathbf{0}\text{.}}

Mathematically the curl of the gradient of any scalar potential is always zero.

## Examples

Gravitational force and Coulomb force are two familiar examples with

F
(

r

)

{\displaystyle F( \mathbf{r} ) }

 being proportional to 1/*r*2 only. An object in such a force field with negative

F
(

r

)

{\displaystyle F( \mathbf{r} ) }

 (corresponding to an attractive force) obeys Kepler's laws of planetary motion.

The force field of a spatial harmonic oscillator is central with

F
(

r

)

{\displaystyle F( \mathbf{r} ) }

 proportional to *r* only and negative.

* A 2D harmonic oscillator with restoring force (x, y) is central and conservative but a harmonic oscillator with restoring force (x, 2y) is not central and yet is conservative.

By Bertrand's theorem, these two,

F
(

r

# )
−
k

/

r
2

{\displaystyle F( \mathbf{r} ) = -k/r^2 }

and *F
(

r

# )
−
k
r

{\displaystyle 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.

## Notes

**a** This article uses the definition of central force given in Taylor. Another common definition (used in *ScienceWorld*) adds the constraint that the force be spherically symmetric, i.e.

F
→

=

F

(

r

# )
F
(

r

)

r

^

.