In electromagnetism, it is often referred to gauges of the electromagnetic field, such as the radiation or Coulomb gauge. As far as I know, the definition of a gauge helps us to redefine the problem in terms of a vector potential and a scalar potential that, since we have some freedom in choosing them, can be chosen in cleverest way it is possible for the given problem.
Here comes my question: is the choice of the gauge a mere mathematical simplification of the given problem? Does this choice have a physical meaning?
My troubles are actually in understanding the physical meaning of this choice of the gauge and what will change if I choose a different gauge.