What causes a force field to be “nonconservative?”

A conservative force field is one in which all that matters is that a particle goes from point A to point B. The time (or otherwise) path involved makes no difference.

Most force fields in physics are conservative (conservation laws of mass, energy, etc.). But in many other applications, the time paths DO matter, meaning that the force field is not "conservative."

What causes a force field to be "non-conservative?" Could you give some examples (probably outside of physics)?

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 I am not sure if this qualifies as a force field, but if there is friction in the system, the force is definitely not conservative – Thomas Rot Jul 9 '12 at 13:24 @ThomasRot: Good point. Almost all physics problems say, "neglect frictional forces." – Tom Au Jul 9 '12 at 13:41 Non-conservative only matters in macroscopic level, in atom level every type of force is conservative. – Shuhao Cao Jul 9 '12 at 14:16 @ShuhaoCaoL: Figured that might be the case. Thanks for the confirmation. – Tom Au Jul 9 '12 at 14:20

1 Answer

A force field $F_i(x)$ is conservative if for every curve $C$ from a point $y_1$ to a point $y_2$, we have $\int\limits_C F_i(x)\mathrm{d}x^i$, so that the energy difference between $y_1$ and $y_2$ is independent of the curve taken from one to the other. Equivalently, the integral around a closed curve must be zero, $\oint\limits_C F_i(x)\mathrm{d}x^i=0$ for every closed curve $C$. Alternatively, we require $\nabla\times F=0$, so that we can write $F=\nabla V$; that is, the curl of the force field is zero so that the force field can be expressed as a divergence. Generalizations of this elementary account to higher dimensions in terms of differential forms are possible.

Although Shuhao Cao's comment that whether a physical theory is macroscopic or microscopic will determine whether the theory is conservative is very often correct, nonetheless phenomenological microscopic theories may find it convenient to include nonconservative force fields. For example, the effect of an externally imposed magnetic field on an otherwise microscopic model may be nonconservative. (see Ron's comment below, which points out that variation of an externally imposed magnetic field over time may be used to give an example of a nonconservative field in 4D. The implication of restriction to 3D that is established by my first paragraph has to be removed.)

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 the effect of an externally imposed magnetic field is never nonconservative, since the vector potential does not affect the energy, only the momentum. The way in which magnetic fields are non-conservative is in time-dependent circumstances, where a changing B makes EMF. – Ron Maimon Jul 9 '12 at 17:28 @Ron Silly me, is obvious. Will edit. – Peter Morgan Jul 9 '12 at 17:54