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?
Added; after reading José Figueroa-O'Farrill's answer I had some thoughts I should add:
I am still by far not an expert in gauge theories. But I think, that in gauge theories, one typically has redundant variables, which have no or at least not a direct physical interpretation. So I would agree that the "physical" dynamics takes place at the quotient in the case that the gauge theory itself has a physical meaning (in particular experimental evidence). Concerning quantization, however, if I am not mistaken, the only known quantum gauge theory which corresponding classical gauge theory which experimental support is quantum electrodynamics. For the other physical relevant quantum gauge theories, I think, the classical counterparts play just the role of auxiliary theories in some sense. In this case I would agree, that on the quantum side only the reduced space has physical meaning, but I think for the corresponding classical one, this seems to be a rather pointless question. So the question remains, why it is physically interesting to study on the classical side the reduced phase space in case of gauge theories. Moreover as José Figueroa-O'Farrill points out, the classical reduced space in most cases too complicated to quantize it directly, one would use some kind of extrinsic quantization as BRST instead. I don't know exactly how the situation for gravitation is. I think one can formulate ART as classical gauge theory. But makes it sense in this case to study the reduced classical phase space for quantization purposes? I guess not.