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I am trying to motivate the role of symmetries in physics. In doing so, I would like to distinguish between a theory's symmetries and the symmetry of a system. The ideas are similar but I am not able to think about them in the proper manner. In the end, a system has to obey the symmetries that the theory exhibits but it can have additional symmetries. Also, a system is usually a specific example in a theory. How to think about this?

For example, a theory like GR described by a Lagrangian has symmetries, but a system like a black hole can have symmetries like rotational symmetry or time translation symmetry. How do we think about these two kinds of symmetries?

Qmechanic
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Khushal
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1 Answers1

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When we consider the evolution of a system there are usually two inputs - the laws which determine the dynamics of the system (in physics, these are often differential equations) and the initial conditions. Even if the laws of the system have a particular symmetry, its evolution does not necessarily share that symmetry because the initial conditions may be asymmetric.

For example, the rules of chess have left-right symmetry, but game positions are asymmetric because the starting position does not share this left-right symmetry.

gandalf61
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