动量映射

在数学，尤其在辛几何中，动量映射是一个与辛流形上的李群的哈密顿作用有关的工具，可用于构造作用的守恒量。动量映射推广了经典的动量和角动量。它在各种辛流形的建立中是一个重要的部分，包括将会在后面讨论的symplectic (Marsden–Weinstein) quotients，以及symplectic cuts和sums。

例子

In the case of a Hamiltonian action of the circle G = U(1)，the Lie algebra dual  is naturally identified with R，and the 动量映射is simply the Hamiltonian function that generates the circle action.
Another classical case occurs when M is the cotangent bundle of R³ and G is the Euclidean group generated by rotations and translations. That is，G is a six-dimensional group，the semidirect product of SO(3) and R³. The six components of the 动量映射are then the three angular momenta and the three linear momenta.

Symplectic quotients

Suppose that the action of a compact Lie group G on the symplectic manifold (M，ω) is Hamiltonian，as defined above，with 动量映射. From the Hamiltonian condition it follows that  is invariant under G.
Assume now that 0 is a regular value of μ and that G acts freely and properly on . Thus  and its quotient  are both manifolds. The quotient inherits a symplectic form from M; that is，there is a unique symplectic form on the quotient whose pullback to  equals the restriction of ω to . Thus the quotient is a symplectic manifold，called the Marsden–Weinstein quotient，symplectic quotient or symplectic reduction of M by G and is denoted . Its dimension equals the dimension of M minus twice the dimension of G.