Chapter 5
Basic Principles of Physics
5.1
Symmetry
Symmetry is one of those concepts that occur in our everyday language and
also in physics. There is some similarity in the two usages, since, as is usually
the case, the physics usage generally grew out of the everyday usage but is
more precise. Let’s start with the general usage. Synonyms for symmetry
are words like balanced or well formed. We most often use the idea in terms
of a work of art. The following 4th century greek statue, Figure 5.1, of a
praying boy is a beautiful work of art. This is attributable to the form and
balance. The figure has an almost exact bilateral, axial reflection, symmetry.
A bilateral symmetry is a well defined mathematical operation on the figure:
Establish a mean central axis and place a mirror to reflect every point on the
object in the plane plane of the mirror. You recover almost the same figure.
In fact a Platonist would attribute the beauty in the piece to the presence
of the mathematical symmetry.
Of course, for this case, the symmetry is
not exact but approximate.
These ideas about symmetry can be generalized and at the same time
made more specific. In art and in physics, the idea is that you perform some
algorithmic or well specified operation to the figure or system of interest. If
you recover the same figure or system then you have a symmetry. Later on
we will get very specific as to the definition of symmetry but the basic idea
that you see here will endure. There is some change that you can make and
if after you make the change you have basically the same thing that you
started with, you say that you have a symmetry. If you recover almost the
same figure or system, you have what is called a slightly broken symmetry
or approximate symmetry.
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CHAPTER 5.
BASIC PRINCIPLES OF PHYSICS
Figure 5.1:
Praying Boy
In art, as it will turn out to be the case in
physics, there is a sense of beauty associated with balanced or symmetric
figures.
This ancient greek statue of a praying boy has an approximate
bilateral symmetry.
The first issue is to understand the idea of making a change. In order to
differentiate the parts of this problem, we will call these changes transfor
mations. There are obviously many transformations that you can perform
both in physics and in art.
Moving the figure to the side is an especially
simple example. The set of operations that are shifting of the figure Is an
example of what is called a translation.
In art, if the figure is the same
after it has been translated, the figure possess translation symmetry; the
transformation is a translation and there is a symmetry if the figure is the
identical to the original. In most cases in art with translation symmetry, the
amount of translation that reproduces the original image is an integer mul
tiple of some fixed amount, see Figure 5.3. This is an example of a discrete
translation symmetry. Our earlier example of bilateral transformations or
mirror images is also an example of a discrete family of transformations.
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 Spring '07
 Anderson
 Physics, College Station, Law of Gravitation, Rescale Oscillator Rescale Oscillator

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