# A simple example of symmetry setting the properties of a Physical System

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)

-
Noether's theorem provides many examples of symmetry leading to conservation law. – Qmechanic Apr 2 '13 at 22:30

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.