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  "body": "\n## Summary\n\nMarsden–Weinstein reduction, also called symplectic reduction, builds a smaller [[symplectic manifold|CAN490EPRT]] from one with a Hamiltonian symmetry.\nOne takes a level set of the [[moment map|ZARM34FZ6Z]] and divides it by the symmetry group.\nIt is the geometric form of removing conserved quantities from a mechanical system.\nIt produces many important spaces, such as complex projective spaces and moduli spaces.\n\n## Explanations\n\n### L1 human\n\nA symmetry of a mechanical system gives conserved quantities, and fixing their values cuts down the phase space.\nThe symmetry still moves points inside this smaller set, so one also identifies points that differ by a symmetry.\nWhat is left is again a phase space, with two dimensions less for each dimension of the symmetry group.\nFor example, fixing the energy of $n + 1$ uncoupled harmonic oscillators with equal frequency gives a sphere $S^{2n+1}$, and dividing by the common phase rotation gives the complex projective space $\\mathbb{CP}^n$.\n",
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