Discretization of the HD, SHD and GHD equations in its Lagrangian form.

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

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

where:

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

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

$latex P_a$ and $latex P_b$

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

In SHD we have:

$latex d$

Finally, the GHD equations are:

$latex d$

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