Are all conservative forces a central force?

If a force is a central force and can be written as $\vec{F}(\vec{r})=f{(r)}\hat{r}$ , then it is a conservative force. But is the converse true? I mean, are all conservative forces a central force? If no, can you please provide explanation?

• Gravity on flat earth surface. Commented Oct 17, 2015 at 12:30
• Any force that admits a potential is conservative. To get a non-central, conservative force, just pick a function $V(r, \theta, \phi)$, such that $\partial_\theta V \ne 0$ or $\partial_\phi V \ne 0$. Then $\vec F = -\nabla V$ will be a conservative, non-central force. One simple example $V=\frac 1 2 k x^2$ leads to the conservative force $\vec F = -\vec e_x kx$ which is obviously not central. Commented Oct 17, 2015 at 12:38
• @SebastianRiese These links say that $\vec F= -kx$ is a central conservative force..... books.google.co.in/… and sfu.ca/~boal/211lecs/211lec14.pdf Commented Oct 17, 2015 at 14:31
• @Aniket Just clicked the sfu.ca link, it talk abouk $\vec F = -k\vec x$, that is a different beast (and I would have written it $\vec F = -k \vec r$ ... using $\vec x$ for positions vectors just generates confusion in my opinion $\vec r$ is avoids this). Commented Oct 17, 2015 at 14:32
• @SebastianRiese What kind of a different beast? Commented Oct 17, 2015 at 14:33