Conservative force

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A conservative force is defined as a force that does not depend on the path taken to increase in potential energy.

1 Informal definition
2 Path independence
3 Mathematical description
4 Nonconservative forces
5 See also

[edit] Informal definition

Informally, a conservative force can be thought of as a force that conserves mechanical energy. Suppose a particle starts at point A, and there is a constant force F acting on it. Then the particle is moved around by other forces, and eventually ends up at A again. Though the particle may still be moving, at that instant when it passes point A again, it has traveled a closed path. If the net work done by F at this point is 0, then F passes the closed path test. Any force that passes the closed path test is classified as a conservative force.

The gravitational force, spring force, magnetic force and electric force (at least in a time-independent magnetic field, see Faraday's law of induction for details) are examples of conservative forces, while friction and air drag are classical examples of non-conservative forces (the energy is transferred to the air as heat and cannot be retrieved).

[edit] Path independence

The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative.

A direct consequence of the closed path test is that the work done by a conservative force on a particle moving between any two points does not depend on the path taken by the particle. Also the work done by a conservative force is equal to the negative of change in potential energy during that process. For a proof of that, let's imagine two paths 1 and 2, both going from point A to point B. The variation of energy for the particle, taking path 1 from A to B and then path 2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2, i.e., the work is independent of the path followed, as long as it goes from A to B.

For example, if a child slides down a frictionless slide, the work done by the gravitational force on the child from the top of the slide to the bottom will be the same no matter what the shape of the slide; it can be straight or it can be a spiral. The amount of work done only depends on the vertical displacement of the child.

[edit] Mathematical description

A force F is called conservative if it meets any of these (equivalent - proof) conditions:

The curl of F is zero:

$\nabla \times \vec{F} = 0. \,$

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

$W = \oint_C \vec{F} \cdot \mathrm{d}\vec r = 0.\,$

The force can be written as the gradient of a potential, $Φ$ :

$\vec{F} = -\nabla \Phi. \,$

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.

[edit] Nonconservative forces

Nonconservative forces arise due to neglected degrees of freedom. For instance, friction may be treated without resorting to the use of nonconservative forces by considering the motion of individual molecules; 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 nonconservative forces are friction and non-elastic material stress.