Comment: This demonstrates how to use the energy principle for practical calculations. We use the same procedure as for energy conservation, but include the work of non-conservative forces. In most cases, we need to know the path, r(t ) taken by the object in order to calculate the work done by the non-conservative forces. This method is therefore often not that practical.

11.5 Potential Energy in Three Dimensions

So far we have restricted ourselves to one-dimensional motions. Let us now introduce the potential energy for a three-dimensional motion and a three-dimensional force.

We call a force F(r) conservative if the work done by the force from a point 0 to a point 1 is independent of the path taken. That is, we call a force F(r) conservative, if (and only if) there is a function U (r) so that:

1

W0 to 1 = F(r) · d r = U (r0) − U (r1) , (11.112)

0

for all (possible) paths between r0 and r1.

It is trivial to show that the work is independent of the path if a function U (r) with the property of (11.112) exists. What about the other way—if the work along all paths are the same, can we prove that there must exist a function U (r)? This is also not difficult, since the function is simply defined as the work integral between the two end-points.

How is the force F(r) and the function U (r) related? Let us look at the work done

Because U (r) = U (x , y, z), we can write:

that is:

We call the function U (r) the potential energy for the force (field) F(r).

Criterion for a conservative force: This means that the criterion for a force to be conservative is a bit stronger in twoand three-dimensions than in one dimension. In one dimension, a force F (x ) is conservative if F (x ) is a function of position alone.

In twoand three-dimensions, it is a necessary condition that the force F(r) is only a function of the position, for the force to be conservative. But it is not a sufficient condition. There are forces F(r) that are only functions of position, but that still are not conservative. In order for the force to be conservative, it must be the gradient of a potential:

From calculus, we know that a force F(r) can be written as a gradient of a function, if and only if, the curl of F(r) is zero everywhere (for all r):

because in this case, the integral of all closed curves is zero.

11.5.1 Example: Constant Gravity in Three Dimensions

Problem: Find the potential energy U (r) for the gravitational force G = −mg j. Solution: We want to find a function U (r) that satisfies:

Let us try the solution we already know from one dimension:

We find the gradient of U :

This means that the graviational force G is the gradient of the potential U = mgy, that U = mgy is a potential energy (function) also in three dimensions, and that the constant gravitational force is conservative.

11.5.2 Example: Gravity in Three Dimensions

Problem: The gravitational force on an object of mass m at position r from an object of mass M in the origin is:

Is this force conservative, and can you find the potential energy for this force? Approach: We know that the force is conservative if the work on the object (of mass m) does not depend on the path. Let us find the work done along a path, and demonstrate that it is only dependent on the displacement and not on the path. Solution: The work done on an object when it is moved along a path r(t ) from r(t0) = r0 to r(t1) = r1 is:

We introduce a common trick for such integrals: We use that d (r · r) = 2r · d r, and therefore that r · d r = d ((1/2)r · r), and write the integral as:

(11.125) This integral does not depend on the path, only on the end-points. The gravitational force is therefore conservative.