Environmental Engineering Reference
In-Depth Information
This is fine so long as one is clear on the distinction between the column vector
V and the actual physical vector
. The former is merely an ordered list of three
numbers which tell us the components of
V
in a particular frame of reference. As
such it is not equal to V which is a different ordered list of numbers. This is of
course not in conflict with the statement that the vector
V
V
is identical in the two
V = V
frames, i.e.
. Notice the subtle notation: we use upright boldface to denote
column vectors and italic boldface to denote actual physical vectors.
Let us see the utility of vectors in action by considering a particular example.
The gravitational attraction between two massive particles of mass m 1 and m 2
located at position vectors
x 1 and
x 2 relative to an origin O leads to the following
equation of motion for particle 1:
m 1 d 2
x 1
d t 2
Gm 1 m 2 x 2 x 1
| x 2 x 1 |
=
3 .
(11.10)
Under rotations of the co-ordinate system, none of the quantities in this equation
change since they are vector or scalar quantities and so the equation holds true in
all frames related to each other by a rotation. Put another way, since we built the
equation using vector and scalar quantities it follows that the equation does not
change its form even if we change reference frame. We say that the laws of physics
are invariant under a (global) change of co-ordinates.
Example 11.1.1 When written in component form, Eq. (11.10) can be written
m 1 d 2 x 1 i
d t 2
x 2 i
x 1 i
=
Gm 1 m 2
x 1 j ) ] 3 / 2 ,
(11.11)
[ (x 2 j
x 1 j )(x 2 j
where x 1 i are the components of
x 1 in T etc. We have again used the convention
(introduced first in Section 8.2) which says that repeated indices are summed over,
i.e. there is a sum over j implied in the denominator. Prove that this equation does
not change its form when expressed in terms of components in T given that T and
T are related by the rotation specified by Eq. (11.6).
R ji x 1 j etc. in the numerator of each
side of Eq. (11.11). The denominator on the right hand side needs special consid-
eration. Clearly it represents the distance between the two particles (raised to the
third power) and this ought to be a scalar quantity. Let us check this. If we define
r j
Solution 11.1.1 We can substitute for x 1 i
=
x 2 j
x 1 j and
d 2
r j r j
then we can write
d 2
R kj r k R lj r l
=
R kj R lj r k r l
=
R kj (R 1 ) jl r k r l
=
δ kl r k r l
=
d 2 .
=
(11.12)
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