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In "Hamiltonian Fluid Dynamics" by P. J. Morrison, the author skipped the steps when doing solving the variation problem from equations 19 - 23. Can you better help me understand what he is doing?

I put a snippet of the paper below. I am mostly stuck on the second terms in 20 and 22. I don't see how the Jacobian and the density get within the derivative term. Also why $ \frac{\partial U}{\partial \rho} $ remains within the derivative because I thought he was doing chain rule. I figured he is holding $ \rho_0 $ constant because he doesn't differentiate $ \rho_0 $.

Snapshot of the part I am stuck at

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Got it. The right way to think of things is that $ q(a, t) $ is a field. Then we need to take field variations. With, $ \frac{\partial \rho_0}{\partial a_\mu} = 0 $ and $ a_0 = t $.

\begin{align} \delta S &= \int dt\ d^3a\ \left[ \rho_0 \dot{q}_i \delta \dot{q}_i - \rho_0 \left( \frac{\partial U}{\partial \rho} \delta \rho + \frac{\partial U}{\partial s} \delta s \right) \right] \\ &= \int dt\ d^3a\ \left[ -\rho_0 \ddot{q}_i \delta q_i - \rho_0^2 \frac{\partial U}{\partial \rho} \frac{\partial \frac{1}{\mathcal{J}}}{\partial \frac{\partial q^k}{\partial a^j}} \delta \frac{\partial q^k}{\partial a^j} \right] \\ &= \int dt\ d^3a\ \left[ -\rho_0 \ddot{q}_i \delta q_i + \frac{\rho_0^2}{\mathcal{J}^2} \frac{\partial U}{\partial \rho} \frac{\partial \mathcal{J}}{\partial \frac{\partial q^k}{\partial a^j}} \delta \frac{\partial q^k}{\partial a^j} \right] \ \end{align}

From Equation 13, \begin{align} \frac{\partial}{\partial \frac{\partial q^k}{\partial a^j}} \left[ \frac{\partial q^k}{\partial a^j} A^i_k \right] &= \frac{\partial}{\partial \frac{\partial q^k}{\partial a^j}} \left[ \mathcal{J} \delta^i_j \right] \\ A^j_k &= \frac{\partial \mathcal{J}}{\partial \frac{\partial q^k}{\partial a^j}} \\ \end{align}

Thus, \begin{align} \delta S &= \int dt\ d^3a\ \left[ -\rho_0 \ddot{q}_i \delta q_i + \frac{\rho_0^2}{\mathcal{J}^2} \frac{\partial U}{\partial \rho} A^j_k\ \delta \frac{\partial q^k}{\partial a^j} \right] \\ &= \int dt\ d^3a\ \left[ -\rho_0 \ddot{q}_k - A^j_k \frac{\partial}{\partial a^j} \left( \frac{\rho_0^2}{\mathcal{J}^2} \frac{\partial U}{\partial \rho} \right) \right] \delta q_k \\ 0 &= \rho_0 \ddot{q}_k + A^j_k \frac{\partial}{\partial a^j} \left( \frac{\rho_0^2}{\mathcal{J}^2} \frac{\partial U}{\partial \rho} \right) \end{align}

With Equation 21,

\begin{align} 0 &= \rho_0 \ddot{q}_k + A^j_k \frac{\partial}{\partial a^j} \left( \frac{\rho_0^2}{\mathcal{J}^2} \frac{\partial U}{\partial \rho} \right) \\ &= \rho \ddot{q}_k + \frac{\partial p}{\partial q_k} \\ &= \rho \frac{d}{dt} v + \nabla p \\ &= \rho \left( \frac{\partial v}{\partial t} + v \cdot \nabla v \right) + \nabla p \end{align}

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