From Wikipedia, the free encyclopedia

A conservative force is a force that does zero net work on a particle that travels along any closed path in an isolated system.

Informally, a conservative force can be thought of as a force that conserves mechanical energy. Say a particle is moving in the positive x direction with some initial kinetic energy Ki. Let a constant conservative force act on the particle in the negative x direction to decelerate it. Eventually, the particle will come to rest (Its kinetic energy will be zero) after a certain displacement. As the force continues to act, the particle then accelerates (in the negative direction) and returns back to its starting point. Since the only force acting on the particle as it travels this closed path is a conservative one, the particle will have the same kinetic energy Ki as it had initially, and thus the net work done by the force through this path is zero.

The situation just described is known as the closed path test. Any force that passes the closed path test is classified as a conservative force.

It is known from experiment that the gravitational force is a conservative force, as well as the spring force. Other examples include the electric force, and the magnetic force (in a time-independent electric field).

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. 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.

It is easy to prove the path independence of conservative forces. Say we have two points, a and b. Let us move a particle from a to b along path 1, and then back again from b to a along a different path which we will call path 2. The entire trip is obviously a closed path since we are back at point a.

Denote the work done by the force from a to b along path 1 as W_ab1. Denote the work done from b to a along path 2 as W_ba2. Since we have a closed path and a conservative force, the net work is equal to zero, or

W_ab1 + W_ba2 = 0

which implies

W_ab1 = -W_ba2 (1)

Now let us move the particle from a to b along path 2 and then back from b to a again along path 2. Therefore

W_ab2 + W_ba2 = 0

which implies

W_ab2 = -W_ba2 (2)

Substituting W_ab2 from equation (2) for -W_ba2 in equation (1), we get

W_ab1 = W_ab2

In other words, the work done on the particle through path 1 equals the work done through path 2. Thus the work does not depend on the path taken.

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

• The curl of F is zero:

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

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

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 nonconservative forces are friction and non-elastic material stress.

• Conservative vector field

This classical mechanics-related article is a stub. You can help Wikipedia by expanding it.