Central force

Central force
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Central_force".

In classical mechanics, a central force 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: ^[1]

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

where F is the force, r is the position vector, |r| is its length and r̂ is the corresponding unit vector, r̂ = r/|r|, and F is a scalar function, F: [0, +∞) → R.

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:

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

(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:

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

(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:

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

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.

References

^ Eric W. Weisstein (1996–2007). "Central Force". ScienceWorld. Wolfram Research. Retrieved on 2008-08-18.

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