Conservative Force - Mathematical Description

Mathematical Description

A force field F, defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector field if it meets any of these three equivalent conditions:

1. The curl of F is zero:

2. There is zero net work (W) done by the force when moving a particle through a trajectory that starts and ends in the same place:

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

Proof that these three conditions are equivalent when F is a force field
1 implies 2: Let C be any simple closed path (i.e., a path that starts and ends at the same point and has no self-intersections), and consider a surface S of which C is the boundary. Then Stokes' theorem says that 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 and define a function 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 So statement 2 implies statement 3 (see full proof). 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, Q.E.D.. (The equivalence of 1 and 3 is also known as (one aspect of) Helmholtz's theorem.)

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

1 implies 2: Let C be any simple closed path (i.e., a path that starts and ends at the same point and has no self-intersections), and consider a surface S of which C is the boundary. Then Stokes' theorem says that

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 and define a function

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

So statement 2 implies statement 3 (see full proof).

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, Q.E.D..

(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, while others do not. 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.

Read more about this topic: Conservative Force

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