辛几何
$$\frac{\partial H}{\partial p_i}=\dot{x_i}\qquad \frac{\partial H}{\partial x_i}=-\dot{p_i}\\
\quad\\
x=(p_1,\cdots,p_n,x_1,\cdots,x_n)\\
\quad\\
\nabla H=(\frac{\partial H}{\partial p_1},\cdots,\frac{\partial H}{\partial p_n},\frac{\partial H}{\partial x_1},\cdots,\frac{\partial H}{\partial x_n})\\
\quad\\
J=\begin{pmatrix}O&-I_n\\I_n&O\end{pmatrix}\\
\quad\\
J\cdot\nabla H=\dot{x}\\
\qquad\\
\{f,g\}=\sum_{k=1}^n[\frac{\partial f}{\partial p_k}\frac{\partial g}{\partial x_k}-\frac{\partial f}{\partial x_k}\frac{\partial g}{\partial p_k}]\\
\qquad\\
\dot{x}=\{H,x\}
$$
$(M,\omega)$，$M$光滑，$\omega\in \Omega^2(M)$非退化，$\dd \omega=0$
由于$\omega$非退化，故辛流形为偶数维，若$M$为$2n$维流形，则$\omega^n$为其上体积形式
若存在$\omega=\dd \theta$，则称$\theta$为$\text{Liouville 1-form}$，称$i(Z)\omega=\theta$对应的$Z$为
$\text{Liouville vector-field}$
若存在伪黎曼度规$g(X,Y)=g(JX,JY)=\omega(JX,Y)$与$h=g-i\omega$，则称$h$为$\text{pesudo K}\ddot{a}\text{hler-form}$，若$g$正定，则称$h$为$\text{K}\ddot{a}\text{hler-form}$
$\displaystyle\omega_{\mathbb{R}^{2n}}=\sum_{i=1}^n\dd p_i\wedge \dd x_i$与$\displaystyle\omega_{FS}=i\partial\overline{\partial}\ln\left({\sum_{i=1}^n|z_i|^2}\right)$均为辛形式，后者称为
$\text{Fubini-Study form}$，$(\mathbb{C}P^n,\omega_{FS})$为$\text{K}\ddot{a}\text{hler}$流形
若两个辛流形$(M,\omega_M)$，$(N,\omega_N)$之间的微分同胚$\psi:M\to N$满足以下条件，则称其为辛同胚
$$\psi^* \omega_N=\omega_M$$
辛同胚存在的必要条件为$\dim M\leq \dim N$
称生成单参辛同胚群$\psi_t$的矢量场$X$为辛矢量场，显然有$L_X\omega=0$
称$Sp(n,\mathbb{R})=\{g\in SL(2n,\mathbb{R})|g^T J g=J\}$为辛群
若$(M,\omega_t)$配备了一族辛形式$\omega_t$，$ t\in [0,1]$，则存在一族辛同胚$\varphi_t:M\to M$，
$t\in [0,1]$，使得
$$\varphi_0=1\qquad\varphi_t^* \omega_t=\omega_0$$
若$(M,\omega)$为$2n$维辛流形，则$\forall p\in M$，存在含$p$的坐标卡
$(U,x_1.\cdots,x_n,p_1,\cdots,p_n)$，在$U$上有
$$\omega=\sum_{i=1}^n \dd p_i\wedge\dd x_i$$
局部坐标$(x_1.\cdots,x_n,p_1,\cdots,p_n)$称为达布坐标
若辛流形$(M,\omega)$的子流形$L\subset M$满足$\omega|_L=0$与$\dim L=\displaystyle\frac{1}{2} \dim M$，则称$L$为$(M,\omega)$的拉格朗日子流形
拉格朗日子流形相当于辛几何中的刚体，对其进行微扰则其将不再$\text{Lagrangian}$
辛流形$(M,\omega)$的每个拉格朗日子流形$L$均含有辛同胚于余切丛$T^*L$的邻域
对辛流形上的任意光滑函数$H\in C^\infty(M)$，由下式定义哈密顿矢量场$X_H$
$$i(X_H)\omega=-\dd H$$
定义泊松括号$\{f,g\}=\omega(X_f,X_g)=-i(X_g)\dd f=-X_g(f)=X_f(g)$
对于配备有正则辛结构$\displaystyle\dd \omega=\sum_{i=1}^n \dd p_i\wedge\dd x_i$的相空间$\mathbb{R}^{2n}$，
$$\displaystyle X_f=\sum_{i=1}^n f_{pi}\frac{\partial}{\partial p_i}+\sum_{i=1}^nf_{xi}\frac{\partial}{\partial x_i}\\
\qquad\\
\displaystyle i(X_f)\omega=\sum_{i=1}^n f_{pi}\dd x_i-\sum_{i=1}^nf_{xi}\dd p_i=-\dd f\\
\qquad\\
\displaystyle f_{pi}=-\frac{\partial f}{\partial x_i}\qquad f_{xi}=\frac{\partial f}{\partial p_i}\\
\qquad\\
X_f=\sum_{i=1}^n[\frac{\partial f}{\partial p_i}\frac{\partial }{\partial x_i}-\frac{\partial f}{\partial x_i}\frac{\partial }{\partial p_i}]$$
可见泊松括号定义与经典定义一致，其与李括号关系为$[X_f,X_g]=X_{\{f,g\}}$
$$i([X_f,X_g])\omega=i(L_{X_f}X_g)\omega\\
\qquad\\
=L_{X_f}i(X_g)\omega-i(X_g)L_{X_f}\omega\\
\qquad\\
=L_{X_f}i(X_g)\omega-i(X_g)[i(X_f)\dd+\dd i(X_f)]\omega\\
\qquad\\
=L_{X_f}i(X_g)\omega=[i(X_f)\dd+\dd i(X_f)]i(X_g)\omega\\
\qquad\\
=d\omega(X_g,X_f)=i(X_{\{f,g\}})\omega$$
在证明中说明了对于哈密顿矢量场有$L_{X_f}\omega=0$，即哈密顿矢量场一定为辛矢量场
泊松括号具有以下性质
$$\{f,\{g,h\}\}=X_fX_g(h)=[X_f,X_g]h+X_gX_f(h)\\
\quad\\
=X_{\{f,g\}}(h)+X_g\{f,h\}\\
\qquad\\
=-\{h,\{f,g\}\}-\{g,\{h,f\}\}$$
令$\psi_t$为$X_H$生成的流，$\varphi_s$为$X_f$生成的流
$$\frac{\dd f(\psi_t(p))}{\dd t}=X_H(f)=\{H,f\}\\
\qquad\\
\frac{\dd H(\varphi_s(p))}{\dd s}=X_f(H)=\{f,H\}\\
\qquad\\
\frac{\dd H(\varphi_s(p))}{\dd s}=-\frac{\dd f(\psi_t(p))}{\dd t}$$
$H$在每种变换$\varphi_s$下的不变性都对应着一个守恒量$f$
配备有$n$个泊松对易$\{I_i,I_j\}=0$的哈密顿量且$\forall x\in M,\displaystyle\bigwedge_{i=1}^n\dd I_i(x)\neq 0$的$2n$维辛流形$(M,\omega)$称为完全可积系统，作映射$\pi=(I_1,\cdots,I_n):M\to \mathbb{R}^n$，则$\pi$的非奇点像集的原像中的连通紧集为一个拉格朗日子流形且其微分同胚于$T^n$
$$H=\frac{1}{2}\sum_{i=1}^N p_i^2+\sum_{i<j}\frac{g^2}{(x_i-x_j)^2}\\
\qquad\\
\dot{x_i}=\{H,x_i\}=p_i\qquad \dot{p_i}=\{H,p_i\}=\sum_{j\neq i}\frac{2g^2}{(x_i-x_j)^3}\\
\qquad\\
L_{ij}=\left\lbrace\begin{array}[l]
&p_i&i=j\\
\displaystyle\frac{\sqrt{-g^2}}{x_i-x_j}&i\neq j
\end{array}\right.\qquad M_{ij}=\left\lbrace\begin{array}[l]
&\displaystyle\sum_{k\neq i}\frac{\sqrt{-g^2}}{(x_i-x_k)^2}&i=j\\
\displaystyle\frac{-\sqrt{-g^2}}{(x_i-x_j)^2}&i\neq j
\end{array}\right.$$
$$\frac{\dd L}{\dd t}=LM-ML$$
考虑包含$\text{Calogero-Moser system}$在内的更一般的情况，$u,w,y$为偶函数，$v$为奇函数
$$H=\frac{1}{2}\sum_{i=1}^N p_i^2+\sum_{i<j}g^2u(x_i-x_j)\\
\qquad\\
L_{ij}=\left\lbrace\begin{array}[l]
&p_i&i=j\\
\displaystyle\sqrt{-g^2}v(x_i-x_j)&i\neq j
\end{array}\right.\\
\qquad\\
M_{ij}=\left\lbrace\begin{array}[l]
&\displaystyle\sum_{k\neq i}\sqrt{-g^2}w(x_i-x_k)&i=j\\
\displaystyle-\sqrt{-g^2}y(x_i-x_j)&i\neq j
\end{array}\right.$$
利用$\dot{f}=\{H,f\}$
$$\dot{L_{ii}}=-g^2\sum_{k\neq i}u’(x_i-x_k)\\
\qquad\\
\dot{L_{ij}}=\sqrt{-g^2}(p_i-p_j)v’(x_i-x_j)\qquad i\neq j\\
\qquad\\
(LM-ML)_{ii}=2g^2\sum_{k\neq i}v(x_i-x_k)y(x_i-x_k)\\
\qquad\\
(LM-ML)_{ij}=-\sqrt{-g^2}(p_i-p_j)y(x_i-x_j)\\
+g^2\sum_{k\neq i,j}[v(x_i-x_k)y(x_j-x_k)+v(x_j-x_k)y(x_i-x_k)\\
+v(x_i-x_j)(w(x_i-x_k)-w(x_j-x_k))]
$$
若要满足$\text{Lax equation}$，则需
$$u’(x)=-2v(x)y(x)\\
\qquad\\
v’(x)=-y(x)\\
\qquad\\
v(\alpha)y(\beta)+v(\beta)y(\alpha)+v(\alpha-\beta)(w(\alpha)-w(\beta))=0
$$
方程可简化为
$$u(x)=v(x)^2+C\\
\qquad\\
[v(\alpha)v(\beta)]’=v(\alpha-\beta)(w(\alpha)-w(\beta))
$$
考虑具有如下形式的$v(x)$和$w(x)$
$$v(x)=\frac{1}{x}+\sum_{k=0}^{\infty}a_{2k+1}x^{2k+1}\qquad w(x)=\frac{1}{x^2}+\sum_{k=0}^{\infty}b_{2k}x^{2k}$$
令$\beta=0,\alpha=x$有
$$w(x)=\frac{v’’(x)}{2v(x)}+a_1+b_0\\
\qquad [v(\alpha)v(\beta)]’=\frac{1}{2}v(\alpha-\beta)(\frac{v’’(\alpha)}{v(\alpha)}-\frac{v’’(\beta)}{v(\beta)})$$
以下四种函数均为此方程的解
$$v(x)=\frac{1}{x},\frac{a}{\sin(ax)},\frac{a}{\sinh(ax)},\frac{a}{\text{sn}(ax,k)}$$
可知$\text{Calogero-Moser system}$满足$\text{Lax equation}$
令$I_k=\Tr\;{L^k}$，由$\text{Lax equation}$有
$$\frac{\dd I_k}{\dd t}=\text{Tr}(\dot{L}L^{k-1}+L\dot{L}L^{k-2}+\cdots+L^{k-1}\dot{L})\\
\qquad\\
=\text{Tr}((LM-ML)L^{k-1}+L(LM-ML)L^{k-2}+\cdots+L^{k-1}(LM-ML))\\
\qquad\\
=\Tr\;(L^kM-ML^k)=0$$
因此$I^k,k=1,\cdots,n$为$\text{Calogero-Moser system}$的$n$个守恒量
若$\lambda^k,k=1,\cdots,n$为$L$的$n$个特征值，则$I_k$=$\displaystyle\sum_{i=1}^n \lambda_i^k$，反过来将$\lambda$视为$I$的函数，则知$\lambda$也为守恒量，以下证明$\lambda$两两泊松对易，设$\lambda,\varphi$和$\mu,\psi$分别为$L$的两组特征值和归一化特征矢
$$L_{ij}=\left\lbrace\begin{array}[l]
&p_i&i=j\\
\displaystyle\sqrt{-g^2}v(x_i-x_j)&i\neq j
\end{array}\right.$$
由特征方程有
$$
p_k\varphi_k+\sqrt{-g^2}\sum_{l\neq k}v(x_k-x_l)\varphi_l=\lambda\varphi_k\\
\qquad\\
p_k\psi_k+\sqrt{-g^2}\sum_{l\neq k}v(x_k-x_l)\psi_l=\mu\psi_k\\
\qquad\\
\frac{\partial \lambda}{\partial p_k}=\varphi_k\overline{\varphi_k}\\
\qquad\\
\frac{\partial \lambda}{\partial x_k}=\sqrt{-g^2}\sum_{k\neq l}v’(x_k-x_l)(\varphi_l\overline{\varphi_k}-\varphi_k\overline{\varphi_l})\\
\qquad\\
\{\lambda,\mu\}=\sqrt{-g^2}\sum_{k,l=1}^nv’(x_k-x_l)((\psi_l\overline{\psi_k}-\psi_k\overline{\psi_l})\varphi_k\overline{\varphi_k}-(\varphi_l\overline{\varphi_k}-\varphi_k\overline{\varphi_l})\psi_k\overline{\psi_k})$$
记$R_{kl}=\varphi_k\psi_l-\varphi_l\psi_k$，其对$k,l$反称
$$\sqrt{-g^2}\sum_{k\neq l}v(x_k-x_l)R_{lk}=(\lambda-\mu)\varphi_k\psi_k\\
\qquad\\
p_kR_{km}+\sqrt{-g^2}\sum_{k\neq l}v(x_k-x_l)R_{lm}=\lambda\varphi_k\psi_m-\mu\psi_k\varphi_m\\
\qquad\\
\{\lambda,\mu\}=\sqrt{-g^2}\sum_{k,l=1}^nv’(x_k-x_l)(\varphi_k\psi_k\overline{R_{lk}}+\overline{\varphi_k\psi_k}R_{kl})$$