Following \citet{hopkins2013} we use the Lagrangian formalism to determine the
equation of motion for the SPH particles. The derivation found in
\citet{hopkins2013} is somewhat sparse, and so we reproduce it here with more
steps for clarity.

The following derivation is underpinned by the idea of there being two
independent ways of defining the volume associated with a particle in SPH. The
first is the volume associated with the thermodynamical system ($\Delta V$),
from the first law of thermodynamics, and the second being the volume around the
particle in which we conserve an effective neighbour number ($\Delta \tilde{V}$).
These two need not necessarily be linked in any way.

We begin with the SPH lagragian,
\begin{align}
  L(q, \dot{q}) = \frac{1}{2}\sum^N_{i=1} m_i \dot{r}^2_i
    - \sum^N_{i=1} m_i u_i,
  \label{eqn:sph:derivation:sphlagrangian}
\end{align}
and the first law of thermodynamics,
\begin{align}
  \left. \frac{\partial u_i}{\partial q_i} \right|_A =
  -\frac{P_i}{m_i} \frac{\partial \Delta V_i}{\partial q_i},
  \label{eqn:sph:derivation:firstlaw}
\end{align}
where $\mathbf{q} = (\mathbf{r}_1, ..., \mathbf{r}_N, h_i, ..., h_N)$ are the
generalised coordinates of the particles, and the derivative of the internal
energy $u_i$ is taken at fixed entropy $A$. This gives the first concept of
`volume' $\Delta V_i$ that the particles occupy.

As mentioned earlier, the particles also have a volume associated with the
spread of their neighbours and hence their smoothing length. We can write this as
a constraint equation,
\begin{align}
    \phi_i(\mathbf{q}) = \kappa h_i^{n_d}
      \frac{1}{\Delta \tilde{V}} - N_{ngb} = 0,
  \label{eqn:sph:derivation:constraint}
\end{align}
where $N_{ngb}$ is the effecitve neighbour number, $\kappa$ is the volume of
the unit sphere ($\kappa_{3D} = 4\pi/3$), and $n_d$ is the number of spatial
dimensions considered in the problem. It is important to note that $N_{ngb}$
need not be an integer quantity.

\subsection{Lagrange Multipliers}

With these components in hand, we can use the technique of Lagrange multipliers
to enforce the constraint equation, with
\begin{align}
  \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot{q}_i} -
  \frac{\partial L}{\partial q_i} = 
  \sum^{N}_{j=1} \lambda_j \frac{\partial \phi_i}{\partial q_j},
  \label{eqn:sph:derivation:lmsum}
\end{align}
where the $\lambda_j$ are the lagrange multipliers.  We use the second half of
these equations (i.e. $q_i = h_i$) to constrain $\lambda_j$. The differentials
with respect to the smoothing lengths:
\begin{align}
  \frac{\partial L}{\partial \dot{h}_i} = 0, 
  \quad
  \frac{\partial L}{\partial h_i} =
  -\sum^N_{j=1}m_j\frac{\partial u_j}{\partial h_i} =
  -m_i \frac{\partial u_i}{\partial h_i}.
\end{align}
As all terms where $i \neq j = 0$. For the constraint equation we find
\begin{align}
  \frac{\partial \phi_j}{\partial h_i} = 
  \kappa n_d h_j^{n_d -1} \frac{\partial h_j}{\partial h_i}
    \frac{1}{\Delta \tilde{V}_j}
  + \kappa h_j^{n_d} \frac{\partial \Delta \tilde{V}_j}{\partial h_i}
    \frac{1}{\Delta \tilde{V}_j^2},
\end{align}
which clearly reduces to
\begin{align}
  \frac{\partial \phi_j}{\partial h_i} = \kappa
  \left(n_d h_j^{n_d - 1}  \frac{1}{\Delta \tilde{V_j}} +
    h_j^{n_d} \frac{\partial \Delta \tilde{V_j}}{\partial h_i}
    \frac{1}{\Delta \tilde{V_j}^2}
  \right)\delta_{ij},
\end{align}
The first law of thermodynamics gives us an expression for
${\partial u_i}/{\partial h_i}$,
\begin{align}
  \frac{\partial u_i}{\partial h_i} =
  -\frac{P_i}{m_i} \frac{\partial \Delta V_i}{\partial h_i}.
\end{align}
Putting all this together we find that
\begin{align}
  P_i \frac{\partial \Delta V_i}{\partial h_i} = 
  \sum^N_{j=1} \kappa \lambda_j
    & \left( n_d h_j^{n_d -1}  \frac{1}{\Delta \tilde{V_j}}\right. \nonumber\\
    & \left. + ~  h_j^{n_d} \frac{\partial \Delta \tilde{V_j}}{\partial h_i}
      \frac{1}{\Delta \tilde{V_j}^2}
  \right)\delta_{ij}.
\end{align}
This expression can now the rearranged for $\lambda_i$,
\begin{align}
  \lambda_i =
  \left(\frac{P_i}{\kappa} \frac{\Delta \tilde{V}^2_i}{h^{n_d}_i}\right)
    \frac{h_i}{n_d \Delta \tilde{V}_i}
    \frac{\partial \Delta V_i}{\partial h_i}
  \left(1 - \frac{h_i}{n_d \Delta \tilde{V}_i}
    \frac{\partial \Delta \tilde{V}_i}{\partial h_i}
  \right)^{-1}.
\end{align}
In \citet{hopkins2013} the lagrange multiplier is split into two parts such that
\begin{align}
    \lambda_i =
    \left(\frac{P_i}{\kappa}
      \frac{\Delta \tilde{V}^2_i}{h^{n_d}_i}
    \right)\psi_i,
   \label{eqn:sph:derivation:lambda_i}
\end{align}
with
\begin{align}
  \psi_i =
    \frac{h_i}{n_d \Delta \tilde{V}_i}
    \frac{\partial \Delta V_i}{\partial h_i}
    \left(1 - \frac{h_i}{n_d \Delta \tilde{V}_i}
      \frac{\partial \Delta \tilde{V}_i}{\partial h_i}
    \right)^{-1}.
  \label{eqn:sph:derivation:psi_I}
\end{align}

\subsection{Generalised Equation of Motion}

Now that the lagrange multipliers have been constrained, the second half of the
equations in \ref{eqn:sph:derivation:lmsum} ($q_i = \mathbf{r}_i$) can be used
to find the equation of motion. The differentials are given as
\begin{align}
    \frac{\partial L}{\partial \dot{\mathbf{r}}_i} = m_i \dot{\mathbf{r}}_i,
    \quad
    \frac{\partial L}{\partial \mathbf{r}_i} =
      \sum_{j=1}^N P_j \frac{\partial \Delta V_j}{\partial \mathbf{r}_i}.
\end{align}
The differential of \ref{eqn:sph:derivation:constraint} is
\begin{align}
    \frac{\partial \phi_j}{\partial \mathbf{r}_i} =
      - \kappa h_j^{n_d} \frac{\Delta \tilde{V}}{\partial \mathbf{r}_i}
        \frac{1}{\Delta \tilde{V}^2},
\end{align}
as $h_j$ is an \emph{independent coordinate} of a particle from $\mathbf{r}_j$
implying that the partial differential $\partial h_j/\partial \mathbf{r}_i = 0$.
We can substitute these into the equation of motion to find
\begin{align}
  m_i \frac{\mathrm{d} \mathbf{v}_i}{\mathrm{d}t} =  
    \sum^N_{j=1}  P_j \nabla_i \Delta V_j + \lambda_j
      \left(
        \kappa \frac{h_j^{n_d}}{\Delta \tilde{V}_j^2}
      \right)
      \left(
        - \frac{\partial \Delta \tilde{V}_j}{\partial \mathbf{r}_i}
      \right).
\end{align}
This then gives the result from the substitution of the lagrange multipliers, 
\begin{align}
  m_i \frac{\mathrm{d} \mathbf{v}_i}{\mathrm{d}t} =
  \sum^N_{j=1} P_j \nabla_i \Delta \tilde{V}_j \psi_j +
  P_j \nabla_i \Delta V_j.
  \label{eqn:sph:derivation:eom}
\end{align}

Now we need to \emph{specify} what we mean by volumes in terms of SPH
quantities. An example, familiar choice would be $\Delta V_i = m_i/\rho_i$.
Notice that in this definition of volume there is one \emph{particle-carried}
property and one `field' or \emph{smoothed} property. This turns out to be the
case for any smoothed property
\begin{align}
  y_i = \sum^N_{j=1} x_j W_{ij}(h_i)
  \label{eqn:sph:derivation:smoothed}
\end{align}
for a given particle-carried scalar $x_i$. This is also true for the $\Delta
\tilde{V}_i = \tilde{x}_i/\tilde{y}_i$. Using these specifications, we can write
down the volume differentials,
\begin{align}
  \frac{\partial \Delta V_i}{\partial h_i} =
    -\frac{x_i}{y_i^2}\frac{\partial y_i}{\partial h_i},
    \quad
  \nabla_i \Delta V_j = -\frac{x_i}{y_i^2} \nabla_i y_j.
  \label{eqn:sph:derivation:volumediffs}
\end{align}
The spatial differential is fairly straightforward and is given by
\begin{align}
  \nabla_i y_j = \nabla_i W_{ij}(h_j)
    + \delta_{ij}\sum_{k=1}^N \nabla_i W_{ik}(h_i).
  \label{eqn:sph:derivation:nablay}
\end{align}
The differential with respect to the smoothing length is also straightforward
after remembering that $W_{ij}(h_j) = w(|r_{ij}|/h_j)/h_j^{n_d}$. Then,
\begin{align}
  \frac{\partial y_i}{\partial h_i} = -\sum_{j=1}^N \frac{x_j}{h_j}
  \left[
    n_d W_{ij}(h_i) + \frac{|r_{ij}|}{h_i}
    \left. 
      \frac{\partial W(u)}{\partial u}
    \right|_{u=\frac{|r_{ij}|}{h_i}}
  \right].
  \label{eqn:sph:derivation:dydh}
\end{align}
Finally, putting all of the above together, we can reach a
formulation-independent equation of motion for SPH,
\begin{align}
  \frac{\mathrm{d}\vec{v}_i}{\mathrm{d}t} = -\sum_{j=1}^N x_i x_j
  & \left[ \frac{f_{ij}P_i}{y_i^2} \nabla_i W_{ij}(h_i) \right. \nonumber \\
  & \left. + ~ \frac{f_{ji} P_j}{y_j^2}\nabla_i W_{ji}(h_j)\right],
  \label{eqn:sph:derivation:spheom}
\end{align}
with
\begin{align}
  f_{ij} \equiv 1 - 
    \frac{\tilde{x}_j}{x_j}
    \left(
      \frac{h_i}{n_d \tilde{y}_i} \frac{\partial y_i}{\partial h_i}
    \right)
    \left(
      1+\frac{h_i}{n_d \tilde{y}_i} \frac{\partial \tilde{y}_i}{\partial h_i}
    \right)^{-1}
  \label{eqn:sph:derivation:fij}
\end{align}
being the so-called `h-terms' that are essentially correction factors due to the
fact that $h$ is allowed to vary.
