泊松括號

在數學及经典力學中，泊松括號是哈密顿力學重要的運算，在哈密頓表述的動力系統中時間推移的定義起着中心角色。在更一般的情形，泊松括号用来定义一个泊松代数，泊松流形是一个特例。它们都是以西莫恩·德尼·泊松而命名。

取决于时间的向量场演示图。泊松括号是用这个向量场的分量函数定义的。

两个取决于时间的向量场演示图，表示了上一个向量场分量的梯度函数。

两个取决于时间的向量场的叉积示意图，表示了原向量场分量的梯度函数。两个函数的括号是它们的 pq-梯度的叉积的长度。这说明了括号、梯度与叉积的关系；由无穷小梯度向量组成的平行四边形越大，括号越大。

正則坐標

在相空间里，用正則坐標 $(q^{i},p_{j})$ ，两个函数$f(\mathbf {q} ,\ \mathbf {p} ),\ g(\mathbf {q} ,\ \mathbf {p} )$ 的泊松括號具有如下形式：

$\{f,g\}=\sum _{i=1}^{N}\left[{\frac {\partial f}{\partial q^{i}}}{\frac {\partial g}{\partial p_{i}}}-{\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q^{i}}}\right].$

运动方程

哈密顿-雅可比运动方程有一个使用泊松括号的等价表示。这可最直接地用坐标系表示。假设 $f(p,q,t)$ 是流形商一个函数。则我们有

${\frac {\mathrm {d} }{\mathrm {d} t}}f(p,q,t)={\frac {\partial f}{\partial t}}+{\frac {\partial f}{\partial p}}{\frac {\mathrm {d} p}{\mathrm {d} t}}+{\frac {\partial f}{\partial q}}{\frac {\mathrm {d} q}{\mathrm {d} t}}.$

然后，取 $p=p(t)$ 与 $q=q(t)$ 为哈密顿-雅可比方程 ${\dot {q}}={\partial H}/{\partial p}$ 与 ${\dot {p}}=-{\partial H}/{\partial q}$ 的解，我们有

${\frac {\mathrm {d} }{\mathrm {d} t}}f(p,q,t)={\frac {\partial f}{\partial t}}+{\frac {\partial f}{\partial q}}{\frac {\partial H}{\partial p}}-{\frac {\partial f}{\partial p}}{\frac {\partial H}{\partial q}}={\frac {\partial f}{\partial t}}+\{f,H\}.$

从而，辛流形上一个函数 f 的演化可用辛同胚单参数族给出，以时间 t 为参数。丢掉坐标系，我们有

${\frac {\mathrm {d} }{\mathrm {d} t}}f=\left({\frac {\partial }{\partial t}}-\{\,H,\cdot \,\}\right)f.$

算子 $-\{\,H,\cdot \,\}$ 称为刘维尔量。

运动常数

一个可积动力系统可能有能量以外的运动常数。这样的运动常数在泊松括号下将与哈密顿量交换。假设某个函数 $f(p,q)$ 是一个运动常数。这意味着如果 $p(t),q(t)$ 是哈密顿运动方程的一条轨迹或解，则沿着轨迹有 $0={\frac {\mathrm {d} f}{\mathrm {d} t}}$。这也我们有

$0={\frac {\mathrm {d} }{\mathrm {d} t}}f(p,q)={\frac {\partial f}{\partial p}}{\frac {\mathrm {d} p}{\mathrm {d} t}}+{\frac {\partial f}{\partial q}}{\frac {\mathrm {d} q}{\mathrm {d} t}}={\frac {\partial f}{\partial q}}{\frac {\partial H}{\partial p}}-{\frac {\partial f}{\partial p}}{\frac {\partial H}{\partial q}}=\{f,H\}$

这里中间步骤利用运动方程得到。这个方程称为刘维尔方程。刘维尔定理描述了如上给出的一个测度（或相空间上分布函数）的时间演化。

为了使一个哈密顿系统完全可积，所有的运动常数必须互相对合。

定义

Let M be symplectic manifold, that is, a manifold on which there exists a symplectic form: a 2-form $\omega$ which is both closed ($d\omega =0$) and non-degenerate, in the following sense: when viewed as a map $\omega :\xi \in \mathrm {vect} [M]\rightarrow i_{\xi }\omega \in \Lambda ^{1}[M]$, $\omega$ is invertible to obtain ${\tilde {\omega }}:\Lambda ^{1}[M]\rightarrow \mathrm {vect} [M]$. Here $d$ is the exterior derivative operation intrinsic to the manifold structure of M, and $i_{\xi }\theta$ is the interior product or contraction operation, which is equivalent to $\theta (\xi )$ on 1-forms $\theta$.

Using the axioms of the exterior calculus, one can derive:

$i_{[v,w]}\omega =d(i_{v}i_{w}\omega )+i_{v}d(i_{w}\omega )-i_{w}d(i_{v}\omega )-i_{w}i_{v}d\omega$

Here $[v,w]$ denotes the Lie bracket on smooth vector fields, whose properties essentially define the manifold structure of M.

If v is such that $d(i_{v}\omega )=0$, we may call it $\omega$-coclosed (or just coclosed). Similarly, if $i_{v}\omega =df$ for some function f, we may call v $\omega$-coexact (or just coexact). Given that $d\omega =0$, the expression above implies that the Lie bracket of two coclosed vector fields is always a coexact vector field, because when v and w are both coclosed, the only nonzero term in the expression is $d(i_{v}i_{w}\omega )$. And because the exterior derivative obeys $d\circ d=0$, all coexact vector fields are coclosed; so the Lie bracket is closed both on the space of coclosed vector fields and on the subspace within it consisting of the coexact vector fields. In the language of abstract algebra, the coclosed vector fields form a subalgebra of the Lie algebra of smooth vector fields on M, and the coexact vector fields form an algebraic ideal of this subalgebra.

Given the existence of the inverse map ${\tilde {\omega }}$, every smooth real-valued function f on M may be associated with a coexact vector field ${\tilde {\omega }}(df)$. (Two functions are associated with the same vector field if and only if their difference is in the kernel of d, i. e., constant on each connected component of M.) We therefore define the Poisson bracket on $(M,\omega )$, a bilinear operation on differentiable functions, under which the $C^{\infty }$ (smooth) functions form an algebra. It is given by:

$\{f,g\}=i_{{\tilde {\omega }}(df)}dg=-i_{{\tilde {\omega }}(dg)}df=-\{g,f\}$

The skew-symmetry of the Poisson bracket is ensured by the axioms of the exterior calculus and the condition $d\omega =0$. Because the map ${\tilde {\omega }}$ is pointwise linear and skew-symmetric in this sense, some authors associate it with a bivector, which is not an object often encountered in the exterior calculus. In this form it is called the Poisson bivector or the Poisson structure on the symplectic manifold, and the Poisson bracket written simply $\{f,g\}={\tilde {\omega }}(df,dg)$.

The Poisson bracket on smooth functions corresponds to the Lie bracket on coexact vector fields and inherits its properties. It therefore satisfies the Jacobi identity:

$\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0$

The Poisson bracket $\{f,\_\}$ with respect to a particular scalar field f corresponds to the Lie derivative with respect to ${\tilde {\omega }}(df)$. Consequently, it is a derivation; that is, it satisfies Leibniz' law:

$\{f,gh\}=\{f,g\}h+g\{f,h\}$

It is a fundamental property of manifolds that the commutator of the Lie derivative operations with respect to two vector fields is equivalent to the Lie derivative with respect to some vector field, namely, their Lie bracket. The parallel role of the Poisson bracket is apparent from a rearrangement of the Jacobi identity:

$\{f,\{g,h\}\}-\{g,\{f,h\}\}=\{\{f,g\},h\}$

If the Poisson bracket of f and g vanishes ($\{f,g\}=0$), then f and g are said to be in mutual involution, and the operations of taking the Poisson bracket with respect to f and with respect to g commute.

李代數

泊松括號是反交换的，也滿足雅可比恒等式。这使得辛流形上的光滑函数空间成为無限維的李代數，以泊松括號为李括號。相应的李群是辛流形的辛同胚群（也稱為正則變換）。

给定一个可微切丛上的向量场 X，令$P_{X}$为其共轭动量。这个从场到共轭动量的映射为从泊松括號到李括號的李代數反同态：

$\{P_{X},P_{Y}\}=-P_{[X,Y]}\,$。

这个重要结果值得我们给个简短证明。记位形空间的 q 点的向量场 X 为

$X_{q}=\sum _{i}X^{i}(q){\frac {\partial }{\partial q^{i}}}$

其中 $\partial /\partial q^{i}$ 是局部坐标系。X的共轭动量的表达式为

$P_{X}(q,p)=\sum _{i}X^{i}(q)\;p_{i}$

这里 $p_{i}$ 为和坐标共轭的动量函数。这样就有，对相空间的每点 $(q,p)$，

$\{P_{X},P_{Y}\}(q,p)=\sum _{i}\sum _{j}\{X^{i}(q)\;p_{i},Y^{j}(q)\;p_{j}\}$

$=\sum _{ij}p_{i}Y^{j}(q){\frac {\partial X^{i}}{\partial q^{j}}}-p_{j}X^{i}(q){\frac {\partial Y^{j}}{\partial q^{i}}}$

$=-\sum _{i}p_{i}\;[X,Y]^{i}(q)$

$=-P_{[X,Y]}(q,p)\,$

以上对所有 $(q,p)$ 成立，证毕。

另见

• 拉格朗日括号
• Moyal bracket
• Peierls bracket
• 泊松超括号
• 泊松超代数
• 狄拉克括号

参考文献

• Arnold, V. I. Mathematical Methods of Classical Mechanics 2nd ed. New York: Springer. 1989. ISBN 978-0387968902.  引文格式1维护：冗余文本 (link)
• Landau, L. D.; Lifshitz, E. M. Mechanics (Course of Theoretical Physics, vol. I) 3rd ed. Butterworth-Heinemann. 1982. ISBN 978-0750628969.  引文格式1维护：冗余文本 (link)