Central force

In classical mechanics, a central force on an object is a force whose magnitude only depends on the distance r of the object from the origin and is directed along the line joining them:

${\vec {F}}=\mathbf {F} (\mathbf {r} )=F(||\mathbf {r} ||){\hat {\mathbf {r} }}$

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

Equivalently, a force field is central if and only if it is spherically symmetric.

Properties

A central force is a conservative field, that is, it can always be expressed as the negative gradient of a potential:

$\mathbf {F} (\mathbf {r} )=-\mathbf {\nabla } V(\mathbf {r} ){\text{, where }}V(\mathbf {r} )=\int _{|\mathbf {r} |}^{+\infty }F(r)\,\mathrm {d} r$

(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:

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

(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:

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

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.)

As a consequence of being conservative, a central force field is irrotational, that is, its curl is zero, except at the origin:

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

Examples

Gravitational force and Coulomb force are two familiar examples with F(r) being proportional to 1/r². An object in such a force field with negative F (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) proportional to r and negative.

By Bertrand's theorem, these two, F(r) = −k/r² and F(r) = −kr, are the only possible central force field with stable closed orbits.

See also

• Classical central-force problem