Conservative force: Information from Answers.com

Classical mechanics

$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})$
Newton's Second Law

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A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken.^[1] Equivalently, if a particle travels in a closed loop, the net work done (the sum of the force acting along the path multiplied by the distance travelled) by a conservative force is zero.^[2]

It is possible to define a numerical value of potential at every point in space for a conservative force. When an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken.

Gravity is an example of a conservative force, while friction is an example of a non-conservative force.

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

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 for all possible closed paths is classified as a conservative force.

The gravitational force, spring force, magnetic force (according to some definitions, see below) 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 (in both cases, the energy is converted to heat and cannot be retrieved).

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.

Mathematical description

A force field F is called conservative (i.e. it is a conservative vector field) if it meets any of these equivalent conditions:

1. The curl of F is zero:

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

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

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

3. The force can be written as the gradient of a potential,

Φ

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

Proof that these three conditions are equivalent when F is a force field

Main article: Conservative vector field

1 implies 2: Let C be any simple closed path and consider a surface S of which C is the boundary. Then Stokes' theorem says that

$\int_S (\nabla \times \vec{F}) \cdot \mathrm{d}\vec{a} = \oint_C \vec{F} \cdot \mathrm{d}\vec{r}$

If the curl of F is zero the left hand side is zero - therefore statement 2 is true.

2 implies 3: Assume that statement 2 holds. Let c be a simple curve from the origin to a point $x$ and define a function

$\Phi(x) = -\int_c \vec{F} \cdot \mathrm{d}\vec{r}.$

The fact that this function is well-defined (independent of the choice of c) follows from statement 2. Anyway, from the fundamental theorem of calculus, it follows that

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

So statement 2 implies statement 3.

3 implies 1: Finally, assume that the third statement is true. A well-known vector calculus identity states that the curl of the gradient of any function is 0. (See proof.) Therefore, if the third statement is true, then the first statement must be true as well.

This shows that statement 1 implies 2, 2 implies 3 and 3 implies 1, therefore all three are equivalent, and the proof is complete.

(The equivalence of 1 and 3 is also known as (one aspect of) Helmholtz's theorem.)

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 (in a time-independent magnetic field, see Faraday's law), and spring force.

Many forces (particularly those that depend on velocity) are not force fields. In these cases, the above three conditions are not mathematically equivalent. For example, the magnetic force satisfies condition 2 (since the work done by a magnetic field on a charged particle is always zero), but does not satisfy condition 3, and condition 1 is not even defined (the force is not a vector field, so one cannot evaluate its curl). Accordingly, some authors classify the magnetic force as conservative,^[3] while others do not.^[4] It should be emphasized that the magnetic force is an unusual case; most velocity-dependent forces, such as friction, do not satisfy any of the three conditions, and therefore are unambiguously nonconservative.

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.

References

^ HyperPhysics - Conservative force
^ Louis N. Hand, Janet D. Finch (1998). Analytical Mechanics. Cambridge University Press. p. 41. ISBN 0521575729.
^ For example, Mechanics, P.K. Srivastava, 2004, page 94: "In general, a force which depends explicitly upon the velocity of the particle is not conservative. (However, the magnetic force (qv×B) can be included among conservative forces in the sense that it acts perpendicular to velocity and hence work done is always zero". Web link
^ For example, The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory, Rüdiger and Hollerbach, page 178, Web link

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Conservative force

Contents

Informal definition

Path independence

Mathematical description

Nonconservative forces

References