ψ _ := ( q _ p _ ) } 2 f {\displaystyle {\underline {\psi }}:=\left.\left({\begin{aligned}&{\underline {q}}\\&{\underline {p}}\\\end{aligned}}\right)\right\}2f}
Vektor der Ableitungen H _ ψ = ( ∂ q H ∂ p H ) {\displaystyle {{\underline {H}}_{\psi }}=\left({\begin{aligned}&{{\partial }_{q}}H\\&{{\partial }_{p}}H\\\end{aligned}}\right)}
Metrik im Phasenraum J _ _ = ( 0 1 − 1 0 ) {\displaystyle {\underline {\underline {J}}}=\left({\begin{matrix}0&1\\-1&0\\\end{matrix}}\right)}
⟨ x _ , y _ ⟩ = x _ T J _ _ y _ {\displaystyle \left\langle {\underline {x}},{\underline {y}}\right\rangle ={{\underline {x}}^{T}}{\underline {\underline {J}}}{\underline {y}}} Eigenschaften:
M 1 ( q , Q , t ) : p = ∂ q M 1 , P = ∂ Q M 1 ⇒ ∂ p i ∂ q k = ∂ 2 M 1 ∂ Q k ∂ q i = − ∂ P k ∂ q i {\displaystyle {{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
M 2 ( q , P , t ) M 3 ( p , Q , t ) M 4 ( p , P , t ) {\displaystyle {\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
ψ _ → ⏟ M ϕ _ {\displaystyle {\underline {\psi }}\underbrace {\to } _{M}{\underline {\phi }}}
mit
ϕ _ := ( Q _ P _ ) } 2 f {\displaystyle {\underline {\phi }}:=\left.\left({\begin{aligned}&{\underline {Q}}\\&{\underline {P}}\\\end{aligned}}\right)\right\}2f}
M − 1 = J − 1 M T J J = M T J M det ( M ) = 1 {\displaystyle {\begin{aligned}&{{M}^{-1}}={{J}^{-1}}{{M}^{T}}J\\&J={{M}^{T}}JM\\&\det \left(M\right)=1\end{aligned}}}
aus LA folgt
ϕ ˙ _ = M − 1 ψ ˙ _ H _ ϕ = M T H _ ψ {\displaystyle {\begin{aligned}&{\underline {\dot {\phi }}}={{M}^{-1}}{\underline {\dot {\psi }}}\\&{{\underline {H}}_{\phi }}={{M}^{T}}{{\underline {H}}_{\psi }}\end{aligned}}}