- Symmetries and Noether’s Theorem
Symmetries are ubiquitous in nature. We admire the symmetrical and intricate patterns on the wing of a butterfly, the simple and symmetrical ripples formed on the surface of still water when a pebble is thrown and so on. Symmetries are not restricted to visual objects. A piece of music where a phrase repeats periodically is also symmetric and pleasing to the ear. We may define symmetry as a transformative property that makes an object possessing such a symmetry look (or feel, sound, etc.) the same after the transformation. Objects that do not look the same after the transformation are not symmetric under that transformation. For example, a palindrome is symmetric when the last letter is replaced by the first, the second to the last by the second, and so on. But it is not symmetric under other transformations such as the interchange of oddly and evenly located alphabets.