# All central forces are conservative forces, but are all conservative forces central forces?

I have just been introduced to the concept of central forces, and to the fact that they are per definition conservative forces. I have looked up several examples of central forces (gravity, electric, and spring), but they cover just about all the conservative forces I have ever heard about. Are there any conservative forces that are not central?

There must be, because otherwise there would not be any point in having a subcategory for central forces, yet I cannot find any examples anywhere.

• Possible duplicates: physics.stackexchange.com/q/213061/2451 , physics.stackexchange.com/q/38874/2451 and links therein. – Qmechanic Aug 22 '18 at 17:29
• You need to be careful about the phrase "per/by definition". In colloquial English we usually use this phrase to mean "it logically follows from the definition", but in math this usage isn't useful because literally everything "logically follows from the definition(s)". So in math and physics we usually use that phrase to mean "it is a necessary part of the definition." Under this usage, (spherically symmetric) central forces are conservative, but they are not conservative "per definition". – tparker Aug 22 '18 at 17:38
• @tparker, in mathematics one should distinguish between "by definition/axiom" and "by theorem". Although everything follows from the basic set of axioms theorems are not set to be true intentionally. In natural sciences, however, the phrase has even greater importance and different sense, as there are postulates and observations, to be distinguished. – user168013 Aug 22 '18 at 23:58

If $\phi=-xy$, and ${F}=-\nabla \phi=y \hat{i}+x \hat{j}$ is conservative but not central

A constant force is conservative but not central.

For example: $\vec F=F \hat x$

You can check that the curl of this force is $0$, hence it is conservative. Its potential energy function in 3D space would just be $V(x,y,z)=-Fx+V_0$, where $V_0$ is some constant value.

An example of this is the approximation of gravity near the Earth's surface. In this regime the force is assumed to be constant, and we get the same form as above for the potential energy. Of course gravity on larger scales is a central force for, say, planets in orbit around a central star, which is why I gave the general form first.

Another example of a conservative, non-central force is one that is a superposition (sum) of two central forces.

• This is a limit of central forces though where the center goes to infinity, so it's in the closure of the set of nonexamples :) – Ryan Thorngren Aug 22 '18 at 17:30
• If the OP has "just been introduced to the concept of Central Forces" then they probably don't know what the curl is – tparker Aug 22 '18 at 17:30
• @RyanThorngren just because a constant force is the limit of a central force where the center is at infinity doesn't mean a constant force is a non-example. You can have an actual constant force that is not this limit. – Aaron Stevens Aug 22 '18 at 17:52
• @tparker I have posted answers at simpler levels when the OP is at a simpler level and gotten strong responses from others here that I should go into more detail when possible. I also show that there is a potential energy function for someone who doesn't know about the curl of a vector field. – Aaron Stevens Aug 22 '18 at 17:54
• @tparker example in comments of the question physics.stackexchange.com/questions/419626/… – Aaron Stevens Aug 22 '18 at 17:58

The total potential of any distribution of sources of a conservative potential is itself conservative, but not necessarily central.