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Does anybody know of an example were one could derive some important properties of a physical system from a symmetry of said system.

I´m specially looking for simple classical examples, which could serve to illustrate the importance of finding symmetries of a system to non-physicists (high school students or first year undergrads)

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Noether's theorem provides many examples of symmetry leading to conservation law. –  Qmechanic Apr 2 '13 at 22:30
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My favourite was always angular momentum conservation of planetary orbits. The gravitational potential energy $$V(r) = - G\frac{M m}{r}$$ depends only on the radial distance $r$ and not the angle, i.e. it is rotationally symmetric. Therefore, the angular momentum of a planet orbiting the Sun is conserved. This all comes out rather beautifully from the Lagrangian formulation of classical mechanics, but for high school/first year students you might have to work a little harder to prove it. Of course, this is just a specific example of the Noether theorem mentioned by Qmechanic.

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Angular momentum is a good one, but it would be nice to find a more "down to Earth" example (sorry for the pun). Does a spinning top needs to be invariant under any rotation? Or does less symmetric top works? (say invariant under 90 degree rotations - with a square cross section). This looks promissing, I'm trying to understand it: en.wikipedia.org/wiki/Rattleback –  Forever_a_Newcomer Apr 3 '13 at 14:50
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