We try to write the formulas in its most general form and fix

First of all, in the simplest form of the HD equations we have

$\frac{d\boldsymbol{v}_a}{dt} = -\sum_b m_b (\frac{P_a}{\rho_a^2} + \frac{P_b}{\rho_b^2}) \nabla_a W_{ab}$ (momentum)

where:

$\rho_a = \sum_b m_b W_{ab}$ (density),

$\frac{de_a}{dt} = \frac{1}{2} \sum_b m_b (\frac{P_a}{\rho_a^2} + \frac{P_b}{\rho_b^2}) \boldsymbol{v}_{ab} \cdot \nabla_a W_{ab}$ (density energy),

$P_a$ and $P_b$

are obtained from the EoS and $\frac{d\boldsymbol{r}_a}{dt} = \boldsymbol{v}_a$.

In SHD we have:

$d$

Finally, the GHD equations are:

$d$

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