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Unformatted text preview: 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. Lets 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. 167 168 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.mirror images is also an example of a discrete family of transformations....
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- Spring '07