Dictionary

Calculational Dictionary for Mixmaster Ground State

(1)
\begin{align} \frac{\partial S}{\partial \alpha} = 2S = \frac{1}{3} e^{2 \alpha} \left( e^{-4 \beta_{+}} + e^{2 \left( \beta_{+} + \sqrt{3} \beta_{-} \right)} + e^{2 \left( \beta_{+} - \sqrt{3} \beta_{-} \right)} \right) \end{align}
(2)
\begin{align} \frac{\partial S}{\partial \beta_{+}} = \frac{1}{3} e^{2 \alpha} \left( -2 e^{-4 \beta_{+}} + e^{2 \left( \beta_{+} + \sqrt{3} \beta_{-} \right)} + e^{2 \left( \beta_{+} - \sqrt{3} \beta_{-} \right)} \right) \end{align}
(3)
\begin{align} \frac{\partial S}{\partial \beta_{-}} = \frac{1}{3} e^{2 \alpha} \left( \sqrt{3} e^{2 \left( \beta_{+} + \sqrt{3} \beta_{-} \right)} - \sqrt{3} e^{2 \left( \beta_{+} - \sqrt{3} \beta_{-} \right)} \right) \end{align}

In terms of

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