# Difference between symmetry and invariance

I'm wondering what's the real difference between symmetry and invariance in Physics? I believe that sometimes the two words are given the same meaning and some other times they are used in a different way.
To me a symmetry is more of a physical property: if you rotate the experiment-table and you get the same results, the system is symmetric with respect to rotations.

While invariance is a mathematical concept, like a gauge transformation.

Is it right to think about it this way? Any insights?

• Nice question, thanks for asking something I had wanted clarification on myself. – user81619 Aug 28 '15 at 18:21
• Possible duplicate: physics.stackexchange.com/q/7700/2451 and links therein. – Qmechanic Aug 30 '15 at 13:35

In particular, the "invariance under a symmetry transformation" means that an object, like the action $S$, has the same value if all the dynamical variables (coordinates, momenta, fields etc.) are transformed according to the symmetry transformation.
If the laws may be derived from one scalar quantity such as the action $S$, the symmetry of the laws of physics is equivalent to the invariance of this scalar quantity under the transformations.