Relevant Reference Link:
https://plus.google.com/u/0/115467095957398372179/posts/7uAwfpWoP4w
Here is a nice article on the increasing role of symmetry in physics by the Nobel Prize winner David Gross
http://www.pnas.org/content/93/25/14256.full
The article suggested by Arthur above is one of my all-time favourite scientific article. Besides, contrary to what is taught in most introductory physics courses, conversational laws do not "drop" from "the above". Actually there is an underlying symmetry which "naturally" leads to a corresponding conversational law. Example: translational symmetry leads to the law of conservation of linear momentum; rotational symmetry leads to the law of conservation of angular momentum, and so on. Credits to Emmy Noether, what a woman!
https://en.wikipedia.org/wiki/Symmetry_(physics)
In fact, many researchers either try to find new symmetries in a system, or, find a system where symmetry breaking occurs. Such findings have led to Nobel prizes in the past! This ascertains the importance of symmetry in physics, I guess. In my opinion, even though Emmy Noether's contribution is of the same order (if not greater) as that from Einstein and other eminent physicists, she is not well recognized unlike them unfortunately.
Also a latest development on the subject can be found here:
https://phys.org/news/2016-01-space-universal-symmetry.html
Symmetries give rise to conservation laws in Classical Mechanics and conserved quantum numbers in Quantum Mechanics. Conservation laws are used to solve problems (eg: energy conservation or angular momentum conservation is used to solve problems of classical mechanics). In Quantum Mechanics (and Quantum Field Theory) generators of conserved quantities (eg: Hamiltonian operator generates time translation symmetry) are special operators. Quantum states are eigenfunctions of these operators and their eigenvalues can be used to label quantum states. Complete sets of such eigenfunctions form complex Hilbert spaces where much of the subsequent analysis is performed.
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