Symplektische Struktur

${\underline {\psi }}:=\left.\left({\begin{aligned}&{\underline {q}}\\&{\underline {p}}\\\end{aligned}}\right)\right\}2f$

Vektor der Ableitungen

${{\underline {H}}_{\psi }}=\left({\begin{aligned}&{{\partial }_{q}}H\\&{{\partial }_{p}}H\\\end{aligned}}\right)$

Metrik im Phasenraum

${\underline {\underline {J}}}=\left({\begin{matrix}0&1\\-1&0\\\end{matrix}}\right)$

symplektisches Skalarprodukt

$\left\langle {\underline {x}},{\underline {y}}\right\rangle ={{\underline {x}}^{T}}{\underline {\underline {J}}}{\underline {y}}$

Eigenschaften:

• schiefsymmetrisch
• bilinear
• nicht entartet

erzeugende Kanonische Trafo

${{M}_{1}}\left(q,Q,t\right):p={{\partial }_{q}}{{M}_{1}},P={{\partial }_{Q}}{{M}_{1}}\Rightarrow {\frac {\partial {{p}_{i}}}{\partial {{q}_{k}}}}={\frac {{{\partial }^{2}}{{M}_{1}}}{\partial {{Q}_{k}}\partial {{q}_{i}}}}=-{\frac {\partial {{P}_{k}}}{\partial {{q}_{i}}}}$

analog

${\begin{aligned}&{{M}_{2}}\left(q,P,t\right)\\&{{M}_{3}}\left(p,Q,t\right)\\&{{M}_{4}}\left(p,P,t\right)\end{aligned}}$

also Insgesamt

${\underline {\psi }}\underbrace {\to } _{M}{\underline {\phi }}$

mit 

${\underline {\phi }}:=\left.\left({\begin{aligned}&{\underline {Q}}\\&{\underline {P}}\\\end{aligned}}\right)\right\}2f$

Eigenschaften der Trafo

${\begin{aligned}&{{M}^{-1}}={{J}^{-1}}{{M}^{T}}J\\&J={{M}^{T}}JM\\&\det \left(M\right)=1\end{aligned}}$

aus LA folgt

${\begin{aligned}&{\underline {\dot {\phi }}}={{M}^{-1}}{\underline {\dot {\psi }}}\\&{{\underline {H}}_{\phi }}={{M}^{T}}{{\underline {H}}_{\psi }}\end{aligned}}$