I am doing a derivation on some registration problem. One of the substeps of the algorithm is solving the following variational problem
\begin{equation} \begin{array}{rrclcl} \displaystyle \max_{\phi} & -\int_{X} \log(|\det(\nabla \phi_{t})|) d\mu\\ \textrm{s.t.} & \dot{\phi} = v_{t}(\phi_{t}),\\ \end{array} \end{equation} where $\phi_{t}:\mathbb{R}^{2} \times \mathbb{R} \rightarrow \mathbb{R}^{2}$ is a diffeomorphism along the velocity vector field $v_{t}:\mathbb{R}^{2} \times \mathbb{R} \rightarrow \mathbb{R}^{2}$, $X$ represents the entire image dimension on which the vector field is defined, $\mu$ is a Lebesgue measure and $\nabla$ is the spatial gradient of $\phi_{t}, t \in [0, 1], \dot{\phi}= \frac{d \phi}{dt}$.
This gives us the nonholonomic constrained problem's Lagrangian: \begin{equation} F[\lambda, \phi, \dot{\phi}, \nabla \phi] = -\int_{X} \log(|\det(\nabla \phi_{t})|) d\mu - \lambda(t) \cdot (\dot{ \phi} - v_{t}(\phi_{t})), \end{equation} where the final functional to be maximized is \begin{equation} -\int_{0}^{1} \int_{X} \log(|\det(\nabla \phi_{t})|) d\mu - \lambda(t) \cdot (\dot{ \phi} - v_{t}(\phi_{t})) dt, \end{equation} where $\phi(0) = Id$ and $\phi(1) = \phi(0) + \int_{0}^{1}v_{t}(\phi_{t})$. After computing the composite Lagrangian's functional derivative based on the extended version (with Lagrange multipliers) of the Euler-Lagrange equations, it gives me the following \begin{equation} \nabla \cdot (\nabla \phi)^{-T} \cdot\nabla \phi \cdot \det(\nabla \phi) + 2 v_{t} = 0\\ \end{equation} However I am kind of stuck here on how I will be able to get an update on $\phi$ based on the previous equations. I might be missing something, if anyone could help I'd be very thankful!
p.s.: I will keep updating the post with findings (issue fixtures, ideas).