Conservative force

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A conservative force is a force whose work is path-independent. In other words, in moving an object from point A to point B, the total work done is independent of the path that the object took.

A force F is conservative if it meets these equivalent conditions:

The curl of F is zero:

<math>\nabla \times \mathbf{F} = 0. \,</math>

The work, W, is zero for any simple closed path:

<math>W = \oint_C \mathbf{F} \cdot \mathbf{dR} = 0.\,</math>

The force can be written as the gradient of a potential, <math>\Phi</math>:

<math>\mathbf{F} = -\nabla \Phi. \,</math>

Conservative force fields are curl-less as a direct consequence of Helmholtz decomposition. The term conservative force comes from the fact that when a conservative force exists, it conserves mechanical energy. The most familiar conservative forces are gravity, the electric force, and spring force.

Nonconservative forces arise due to neglected degrees of freedom. For instance, friction may be treated without resorting to the use of nonconservative forces by treating heat as kinetic energy; however that means every molecule's motion must be considered rather than handling it through statistical methods. For macroscopic systems the nonconservative approximation is far easier to deal with than millions of degrees of freedom. Examples of non-conservative forces are friction and non-elastic material stress.