The behavior of reductions described by this word on quotient spaces determined by the momentum map is defined by the Marsden–Weinstein (“wine-stine”) theorem. This word describes the creed that “Everything is a Lagrangian (“luh-GRAHN-jee-an”) submanifold.” The Lie (“lee”) derivative of the top exterior power of mathematical objects described by this word is always zero. On R-2n, the canonical choice for a mathematical object described by this word is the sum over i of: “dq-sub-i wedge dp-sub-i”. The cotangent bundle of a manifold always has the structure of this type of manifold. The Liouville–Arnold (“l’YOO-vill-arnold”) Theorem gives the sufficient conditions for a manifold described by this word to have action-angle coordinates. The action of a differential 2-form described by this word defines the Poisson (“pwah-SAWN”) bracket. For 10 points, what word describes the branch of differential geometry that is the natural setting of Hamiltonian mechanics? ■END■
ANSWER: symplectic [accept symplectic geometry, symplectic manifolds, symplectic forms, symplectic creed, symplectic maps, or symplectic reductions; accept symplectomorphisms] (The sentence about Lie derivatives is a formulation of Liouville’s theorem.)
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