Environmental Engineering Reference
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be written as
·
=−
F
d x
d U.
(9.5)
Using the chain rule, we can therefore write
∂U
∂x 1
d x 3
∂U
∂x 2
∂U
∂x 3
·
=−
+
+
F
d x
d x 1
d x 2
∂U
∂x i
=−
d x i .
(9.6)
In the second line we have again made use of the summation convention
introduced in Eq. (8.7). Since the left hand side is equal to F i d x i and the equality
is true for any infinitesimal line element it follows that we must be able to write
the components of the force as
∂U
∂x i
F i =−
(9.7)
which in vector notion is usually written as
F
=−
U(r),
(9.8)
where
is known as the gradient operator defined by
∂x i
e i
.
(9.9)
Thus, if the gravitational field is to be conservative then it follows that it must be
possible to express it as the gradient of a scalar field, as in Eq. (9.8). Put another way,
if we can find a scalar field U(r) whose gradient gives the force acting upon the mass
m then we will have succeeded in showing that the gravitational field is conservative.
It is not too hard to come up with the correct potential energy function. If we
consider
GMm
r
=−
U
(9.10)
then we can go ahead and compute the corresponding components of the force
using Eq. (9.7). Thus we just need to compute
∂U
∂x i =
d U
d r
∂r
∂x i .
(9.11)
If we put the mass M at the origin then r 2
x 1 +
x 2 +
x 3
=
and so
∂r
∂x i =
x i
r .
(9.12)
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