正準変換の不変量
１ 相対積分不変量
$${(q_i,p_i)\rightarrow(Q_i,P_i)}$$の正準変換の母関数を$${W(\{q_i\},\{Q_i\})}$$とすると
$${p_i=\dfrac{\partial W_1}{\partial q_i}}$$(i-1) $${P_i=-\dfrac{\partial W_1}{\partial Q_i},}$$(i-2) $${\mathscr{\overline{H}}=\mathscr{H}+\dfrac{\partial W_1}{\partial t}}$$(i-3)
である。（正準変換の母関数 i のパターン）
いま$${f}$$次元の$${\{q_i\}}$$空間での任意の経路を考え
$${\{q_i\}}$$空間：$${(\{q_i\})\rightarrow(\{q_i+\delta q_i\})}$$
$${\{Q_i\}}$$空間：$${(\{Q_i\})\rightarrow(\{Q_i+\delta Q_i\})}$$
とする。
$${\delta W(\{q_i\},\{Q_i\})=\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial W}{\partial q_i}\delta q_i+\dfrac{\partial W}{\partial Q_i}\delta Q_i\Big)}$$ ← (i-1)(i-2)を代入
$${=\displaystyle\sum_{i=1}^f\Big(p_i\delta q_i-P_i\delta Q_i\Big)}$$
ここで元の点に戻る閉曲線経路を考えると、$${\delta W=0}$$なので
$${\oint\displaystyle\sum_{i=1}^fp_i\delta q_i=\oint\sum_{i=1}^fP_i\delta Q_i}$$ (1)
が成り立つ。
任意の閉曲線経路についての運動量の積分は正準変換の前後で不変である。$${\oint\displaystyle\sum_{i=1}^fp_i\delta q_i}$$を（$${poincar\'e}$$の）相対積分不変量と呼ぶ。
２ 絶対積分不変量
$${2f}$$次元の位相空間$${(\{q_i\},\{p_i\})}$$での２次元の表面を考え、$${q_i=q_i(u,v)}$$、$${p_i=p_i(u,v)}$$とする。このとき
$${\delta q_i\delta p_i=\begin{vmatrix}J\end{vmatrix}\delta u\delta v=\begin{vmatrix}\dfrac{\partial q_i}{\partial u} \dfrac{\partial q_i}{\partial v}\\\dfrac{\partial p_i}{\partial u} \dfrac{\partial p_i}{\partial v}\end{vmatrix}\delta u\delta v=\Big(\dfrac{\partial q_i}{\partial u}\dfrac{\partial p_i}{\partial v}-\dfrac{\partial q_i}{\partial v}\dfrac{\partial p_i}{\partial u}\Big)\delta u\delta v}$$ (2-1)
位相空間$${(\{Q_i\},\{P_i\})}$$でも同様に
$${\delta Q_i\delta P_i=\begin{vmatrix}J\end{vmatrix}\delta u\delta v=\begin{vmatrix}\dfrac{\partial Q_i}{\partial u} \dfrac{\partial Q_i}{\partial v}\\\dfrac{\partial P_i}{\partial u} \dfrac{\partial P_i}{\partial v}\end{vmatrix}\delta u\delta v=\Big(\dfrac{\partial Q_i}{\partial u}\dfrac{\partial P_i}{\partial v}-\dfrac{\partial Q_i}{\partial v}\dfrac{\partial P_i}{\partial u}\Big)\delta u\delta v}$$
(2-2)
である。（→ 重積分､ヤコビアン）
また母関数を$${W(\{q_i\},\{P_i\})}$$とすると
$${p_i=\dfrac{\partial W_2}{\partial q_i}}$$ (ii-2) $${Q_i=\dfrac{\partial W_2}{\partial P_i}}$$ (ii-3) $${\overline{\mathscr{H}}=\mathscr{H}+\dfrac{\partial W_2}{\partial t}}$$ (ii-4)
である。（正準変換の母関数 ii のパターン）
正準変数堆が作る２次元の表面素片は$${\delta q_i\delta p_1}$$であるので、これを$${q_i,p_i}$$について積分し、$${i}$$について和を取る。
$${\displaystyle\sum_{i=1}^f\iint \delta p_i \delta q_i=\iint \displaystyle\sum_{i=1}^f\delta p_i\delta q_i}$$ ← (2-1)を代入
$${=\displaystyle\iint \sum_{i=1}^f\Big(\dfrac{\partial q_i}{\partial u}\dfrac{\partial p_i}{\partial v}-\dfrac{\partial q_i}{\partial v}\dfrac{\partial p_i}{\partial u}\Big)\delta u\delta v}$$ ← (ii-2)を代入
$${=\displaystyle\iint \sum_{i=1}^f\Big\{\dfrac{\partial q_i}{\partial u}\dfrac{\partial }{\partial v}\Big(\dfrac{\partial W}{\partial q_i}\Big)-\dfrac{\partial q_i}{\partial v}\dfrac{\partial}{\partial u}\Big(\dfrac{\partial W}{\partial q_i}\Big)\Big\}\delta u\delta v}$$ ← $${W}$$を全微分
$${= \displaystyle\iint\sum_{i=1}^f\Big[\dfrac{\partial q_i}{\partial u}\dfrac{\partial }{\partial v}\Big\{\dfrac{\partial}{\partial q_i}\sum_{j=1}^f\Big(\dfrac{\partial W}{\partial q_j}\delta q_j+\dfrac{\partial W}{\partial P_j}\delta P_j\Big)\Big\}}$$
$${-\displaystyle\dfrac{\partial q_i}{\partial v}\dfrac{\partial}{\partial u}\Big\{\dfrac{\partial }{\partial q_i}\sum_{j=1}^f\Big(\dfrac{\partial W}{\partial q_j}\delta q_j+\dfrac{\partial W}{\partial P_j}\delta P_j\Big)\Big\}\Big]\delta u\delta v}$$
$${=\displaystyle\iint\sum_{i=1}^f \Big(\underline{\dfrac{\partial^2 W}{\partial q_i\partial q_j}\dfrac{\partial q_i}{\partial u}\dfrac{\partial q_i}{\partial v}}+\dfrac{\partial^2 W}{\partial q_i\partial P_j}\dfrac{\partial P_j}{\partial v}\dfrac{\partial q_i}{\partial u}}$$
$${-\underline{\dfrac{\partial^2 W}{\partial q_j\partial q_i}\dfrac{\partial q_j}{\partial u}\dfrac{\partial q_i}{\partial v}}-\dfrac{\partial^2 W}{\partial q_i\partial P_j}\dfrac{\partial P_j}{\partial u}\dfrac{\partial q_i}{\partial v}\Big)\delta u\delta v}$$
← 下線部を足すと$${=0}$$
$${=\displaystyle\iint\sum_{i=1}^f\sum_{j=1}^f \Big(\dfrac{\partial^2 W}{\partial q_i\partial P_j}\dfrac{\partial P_j}{\partial v}\dfrac{\partial q_i}{\partial u}-\dfrac{\partial^2 W}{\partial q_i\partial P_j}\dfrac{\partial P_j}{\partial u}\dfrac{\partial q_i}{\partial v}\Big)\delta u\delta v}$$
$${=\displaystyle\iint\sum_{i=1}^f\sum_{j=1}^f \Big(\dfrac{\partial^2 W}{\partial q_i\partial P_j}\dfrac{\partial P_j}{\partial v}\dfrac{\partial q_i}{\partial u}+\underline{\dfrac{\partial^2 W}{\partial P_j\partial P_i}\dfrac{\partial P_j}{\partial v}\dfrac{\partial P_i}{\partial u}}}$$
$${-\dfrac{\partial^2 W}{\partial q_i\partial P_j}\dfrac{\partial P_j}{\partial u}\dfrac{\partial q_i}{\partial v}-\underline{\dfrac{\partial^2 W}{\partial P_i\partial P_j}\dfrac{\partial P_i}{\partial v}\dfrac{\partial P_j}{\partial u}}\Big)\delta u\delta v}$$
← 下線部を足すと$${=0}$$
$${=\displaystyle\iint\sum_{i=1}^f\sum_{j=1}^f\Big\{\dfrac{\partial P_j}{\partial v}\Big(\dfrac{\partial^2 W}{\partial q_i\partial P_j}\dfrac{\partial q_i}{\partial u}+\dfrac{\partial^2 W}{\partial P_j\partial P_i}\dfrac{\partial P_i}{\partial u}\Big)}$$
$${-\dfrac{\partial P_j}{\partial u}\Big(\dfrac{\partial^2 W}{\partial q_i\partial P_j}\dfrac{\partial q_i}{\partial v}+\dfrac{\partial^2 W}{\partial P_i\partial P_j}\dfrac{\partial P_i}{\partial v}\Big)\Big\}\delta u\delta v}$$
$${=\displaystyle\iint\sum_{i=1}^f\sum_{j=1}^f\Big\{\dfrac{\partial P_j}{\partial v}\dfrac{\partial}{\partial u}\dfrac{\partial}{\partial P_j}\Big(\dfrac{\partial W}{\partial q_i}\delta q_i+\dfrac{\partial W}{\partial P_i}\delta P_i\Big)}$$
$${-\dfrac{\partial P_j}{\partial u}\dfrac{\partial}{\partial v}\dfrac{\partial}{\partial P_j}\Big(\dfrac{\partial W}{\partial q_i}\delta q_i+\dfrac{\partial W}{\partial P_i}\delta P_i\Big)\Big\}\delta u\delta v}$$
$${=\displaystyle\iint\sum_{j=1}^f\Big\{\dfrac{\partial P_j}{\partial v}\dfrac{\partial}{\partial u}\dfrac{\partial}{\partial P_j}\underline{\sum_{i=1}^f\Big(\dfrac{\partial W}{\partial q_i}\delta q_i+\dfrac{\partial W}{\partial P_i}\delta P_i\Big)}}$$
$${-\displaystyle\dfrac{\partial P_j}{\partial u}\dfrac{\partial}{\partial v}\dfrac{\partial}{\partial P_j}\underline{\sum_{i=1}^f\Big(\dfrac{\partial W}{\partial q_i}\delta q_i+\dfrac{\partial W}{\partial P_i}\delta P_i\Big)}\Big\}\delta u\delta v}$$
← $${W=W(\{q_i\},\{P_i\})}$$なので、下線部$${=W}$$
$${=\displaystyle\iint\sum_{j=1}^f\Big\{\dfrac{\partial P_j}{\partial v}\dfrac{\partial}{\partial u}\Big(\dfrac{\partial W}{\partial P_j}\Big)-\displaystyle\dfrac{\partial P_j}{\partial u}\dfrac{\partial}{\partial v}\Big(\dfrac{\partial W}{\partial P_j}\Big)\Big\}\delta u\delta v}$$
← (ii-3)より $${\dfrac{\partial W}{\partial P_j}=Q_j}$$
$${=\displaystyle\iint\sum_{j=1}^f\Big(\dfrac{\partial P_j}{\partial v}\dfrac{\partial Q_j}{\partial u}-\displaystyle\dfrac{\partial P_j}{\partial u}\dfrac{\partial Q_j}{\partial v}\Big)\delta u\delta v}$$ ←$${j}$$を$${i}$$に入れ替える
$${=\displaystyle\iint\sum_{i=1}^f\Big(\dfrac{\partial P_i}{\partial v}\dfrac{\partial Q_i}{\partial u}-\displaystyle\dfrac{\partial P_i}{\partial u}\dfrac{\partial Q_i}{\partial v}\Big)\delta u\delta v}$$ ← (2-2)より
$${=\displaystyle\iint\sum_{i=1}^f\delta P_i\delta Q_i}$$
よって
$${\displaystyle\iint\sum_{i=1}^f\delta p_i\delta q_i=\iint\sum_{i=1}^f\delta P_i\delta Q_i}$$ (2-3)
（強引に左辺を変形して右辺に導いたが、左辺＝ .. ＝〇､右辺＝ .. ＝〇。
よって左辺＝右辺でよかった。ふぅ。。）
正準変換$${(q_i,p_i)\rightarrow (Q_i,P_i)}$$において、位相空間での２次元の表面素片の積分は不変である。
この式の両辺に
$${\displaystyle\iint\sum_{j=1}^f\delta p_j\delta q_j=\iint\sum_{i=1}^f\delta P_j\delta Q_j}$$
を掛けあわせると
$${\displaystyle\iint\iint\sum_{i=1}^f\sum_{j=1}^f\delta p_i\delta q_i\delta p_j\delta q_j=\iint\iint\sum_{i=1}^f\sum_{j=1}^f\delta p_i\delta q_i\delta p_j\delta q_j}$$
となり、４次元積分も不変である。
この掛けあわせを$${f}$$回くり返すと
$${\displaystyle\iint\cdots\iint\sum_{i_1=1}^f\cdots\sum_{i_f=1}^f\delta p_{i_1}\cdots\delta p_{i_f}\delta q_{i_1}\cdots\delta q_{i_f}}$$
$${=\displaystyle\iint\cdots\iint\sum_{i_1=1}^f\cdots\sum_{i_f=1}^f\delta P_{i_1}\cdots\delta P_{i_f}\delta Q_{i_1}\cdots\delta Q_{i_f}}$$ (2-4)
この積分は$${2f}$$次元の位相空間の体積であり、Liouvilleの定理に相当する。（→ Liouville の定理）