Datasets:
vnpthack
/
vnpt_data

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
0
6.73k
operation and it appears exactly the same after the operation. Eor instance, If
we look at a silhouette of a vase that is left-and-right symmetrical, then turn it
1802 around the vertical axis, it looks the same. We shall adopt the definition
of symmetry in Weyl's more general form, and in that form we shall discuss
symmetry of physical laws.
Suppose we bưild a complex machine in a certain place, with a lot of compli-
cated interactions, and balls bouneing around with forces between them, and so
on. Now suppose we build exactly the same kind of equipment at some other
place, matching part by part, with the same dimensions and the same orientation,
everything the same only displaced laterally by some distance. Khen, if we start
the two machines in the same initial circumstances, in exact correspondence, we
ask: will one machine behave exactly the same as the other? WIHI ít follow all the
motions in exact parallelism? Of course the answer may well be øø, because 1Í we
choose the wrong place for our machine it might be inside a wall and interferences
from the wall would make the machine not work.
AII of our ideas in physics require a certain amount of common sense in their
application; they are not purely mathematical or abstract ideas. We have to
understand what we mean when we say that the phenomena are the same when
we move the apparatus to a new position. We mean that we move everything
that we believe is relevant; 1f the phenomenon is not the same, we suggest that
--- Trang 213 ---
something relevant has not been moved, and we proceed to look for it. TỶ we
never fñnd it, then we claim that the laws of physics do not have this symmetry.
Ôn the other hand, we may find it—we expect to fnd it—ïf the laws of physics
do have this symmetry; looking around, we may discover, for instance, that the
wall is pushing on the apparatus. The basic question is, if we defñne things well
enough, If all the essential forces are included inside the apparatus, ïf all the
relevant parts are moved from one place to another, wiïll the laws be the same?
WIII the machinery work the same way?
Tt is clear that what we want to do is to move all the equipment and essenfial
Iinuences, but not cuerwthzng in the world—planets, stars, and all—for If we
do that, we have the same phenomenon again for the trivial reason that we are
ripht back where we started. No, we cannot move cuerwth”ng. But it turns out
in practice that with a certain amount of intelligence about what to move, the
machinery will work. In other words, if we do not go inside a wall, If we know
the origin of the outside forces, and arrange that those are moved too, then the
machinery 6 work the same in one location as in another.
11-2 Translations
We shall limit our analysis to just mechanics, for which we now have sufficient
knowledge. In previous chapters we have seen that the laws of mechanics can be
summarized by a set of three equations for each particle:
m(d°+/dt?) = F„, m(d®u/dt2) = Fụ, m(dÊz/d12) = F;. (11.1)
Now this means that there exists a way tO measure ø, ụ, and z on three perpen-
dicular axes, and the forces along those directions, such that these laws are true.
These must be measured from some origin, but œhere do t0e pu‡ the origin? All
that Newton would tell us at fñrst is that there ¡s some place that we can measure
from, perhaps the center of the universe, such that these laws are correct. But we
can show immediately that we can never ñnd the center, because if we use some
other origin it would make no diference. In other words, suppose that there are
two people—Joe, who has an origin in one place, and Moe, who has a parallel
system whose origin is somewhere else (Eig. II-I). Ñow when Joe measures the
location of the point in space, he fnds it at #z, , and z (we shall usually leave z
out because it is too confusing to draw in a picture). Moe, on the other hand,
when measuring the same point, will obtain a diferent + (in order to distinguish
--- Trang 214 ---
JOE_ |MOE
x x xí
Fig. 11-1. Two parallel coordinate systems.
it, we will call it +), and in principle a diferent , although in our example they
are numerically equal. So we have
+ =#— d, =ụ, z'=z. (11.2)
Now in order to complete our analysis we must know what Moe would obtain for
the forces. The force is supposed to act along some line, and by the force in the
z-direction we mean the part of the total which is in the z-direction, which is
the magnitude of the force times this cosine of its angle with the zø-axis. Now
we see that Moe would use exactly the same proJection as Joe would use, so we
have a set of equations
Hạ — Fựạ, Tự = Đụ, đà. —= F). (11.3)
'These would be the relationships between quantities as seen by jJoe and Moe.
The question 1s, if Joe knows Newton”s laws, and If Moe tries to write
down Newton's laws, will they also be correct for hữm? Does it make any
diference rom which origin we measure the points? In other words, assuming
that equations (11.1) are true, and the Bqs. (11.2) and (11.3) gïve the relationship
of the measurements, is iW or is it not true that
(a) m(d®z/di?) = F„„,
(b) m(d2y//4”) = Fạ, (114)
(c)_ m(d2z'/di?) = F„.?
In order to test these equations we shall diferentiate the formula for øˆ bwice.
First of all
dd ( ) d> — da
———= (#—d)=————..
dt dt dt — dt
--- Trang 215 ---
NÑow we shall assume that Moe's origin is fxed (not moving) relative to Joe”s;
therefore ø is a constant and đa/dt = 0, so we fnd that
da /dt = da/dt
and therefore
d2z' /dt? = d°+/d);
therefore we know that Eq. (11.4a) becomes
m(d°+/dt?) = Fị›.
(W© also suppose that the masses measured by Joe and Moe are equal.) Thus
the acceleration times the mass is the same as the other fellow's. We have also
found the formula for F7, for, substituting from Ead. (11.1), we find that
Từ —= Fụ.
Therefore the laws as seen by Moe appear the same; he can write Newton's
laws too, with diferent coordinates, and they will still be right. That means that
there is no unique way to defne the origin of the world, because the laws will
appear the same, from whatever position they are observed.
'This 1s also true: ïf there is a piece of equipment in one place with a certain
kind of machinery in it, the same equipment in another place will behave in the