Environmental Engineering Reference
In-Depth Information
relative to the origin in
T
,i.e.wemustwrite
x
=
x
−
R
. This may look odd,
for we have previously been stressing that vectors remain unchanged as we move
from frame to frame. However, a position vector relative to one origin is not the
same as the position vector representing the same point but relative to a different
origin, that is why
x
=
x
. The equation of motion for particle 1 now becomes
m
1
d
2
(
x
1
−
R
x
2
−
R
x
1
−
R
)
(
)
−
(
)
=
Gm
1
m
2
(11.24)
(
)
d
t
2
3
x
2
−
R
x
1
−
R
)
−
(
and since
R
is constant this reduces to
d
2
x
1
d
t
2
Gm
1
m
2
x
2
−
x
1
=
m
1
3
.
(11.25)
x
2
−
x
1
Thus the form of the equation is once again unchanged. It was not automatic
that this form invariance should occur, in particular it was important that we had
the opportunity to differentiate the vector
on the left hand side. If translational
symmetry is a good symmetry of Nature then we should require all the laws of
physics to possess the same form invariance as we have just discovered for the
law of gravitation.
R
11.1.3 Galilean symmetry
In the previous subsection we showed how vectors and scalars are the building
blocks which ensure that the mathematical expression of the laws of physics accord
with the fact that Nature does not care how we choose to set up our system of
co-ordinates. It is very natural to ask if there are any other symmetries of Nature
which constrain the form of physical laws in analogy to the way that co-ordinate
invariance constrains us to build the laws of physics using vectors and scalars. Of
course we immediately know of one such symmetry from Part II: the principle of
special relativity which states that physics looks the same in all inertial frames.
It was Einstein who elevated Galileo's observation that there appears to be no
experiment able to ascertain whether an object is at rest or moving with uniform
velocity into a fundamental symmetry of Nature. In its Galilean form the principle
of relativity would say that the laws of physics should take the same form in inertial
frames
S
and
S
where
x
=
x
−
V
t.
(11.26)
We might think of the situation as a translation (Figure 11.2) but where the trans-
lational vector
t
. Clearly it is a significant
additional restriction on any physical law that it should be in accord with the
relativity principle.
As a specific example, let us return once again to the two masses interacting
gravitationally. Eq. (11.24) still holds true but now
R
depends linearly on time, i.e.
R
=
V
R
is not a constant vector.
