Do you think that symplectic reduction (Marsden Weinstein reduction) is interesting from a physical point of view? If so, why? Does it give you some new physical insights?
There are some possible answers I often heard of, but I don't really understand it. Perhaps you could comment these points. Explain and illustrate why they are good reasons or if not, explain why they are nonsense:
Symplectic reduction is interesting because "it simplifies" the system under consideration because you exploit symmetries to eliminate some redundant degrees of freedom. I do not really understand what's the point here because in general reduction leads to a more complicated geometry. (Or even to singular spaces if you consider more general reduction settings).
It's interesting because it is a toy model for gauge theories.
It's interesting because if you want to "quantize" a system, from a conceptual point of view, one should start from the reduced system, from the "real" phase space. I don't see why one should do this for nonrelativistic quantum systems. Even for gauge theories I don't get the point, because the usual procedure is quantize the unreduced system (via gauge fixing), isn't it?
If there are points which make symplectic reduction interesting from a physical point of view, are there physical reasons why one should study reduction by stages?