Updated Lagrange formulation

I understand, that one has to specify the kernel of a boundary value problem and the FE discretization is done on the provided "initial" mesh. In continuum mechanics considering large deformations it is also possible to consider the weak form in the current configuration. To that end the discretization and FE-solve must be done successively on an updated mesh. (see e.g. [example55](https://github.com/kinnala/scikit-fem/blob/master/docs/examples/ex55.py) of the scikit-fem library)
Starting point is the week form in the current configuration
```math
0 = \int_{\omega} \frac{1}{J}\boldsymbol{\tau}(\vec{u}) : \vec{\nabla}(\vec{v}) \, dv
```
with the Kirchhoff stress for a Neo-Hookean solid
```math
\boldsymbol{\tau} = 2 \frac{\partial \psi}{\partial \boldsymbol{b}}\cdot\boldsymbol{b}
= \mu \left[\boldsymbol{b}-\boldsymbol{1}\right]+\lambda\ln(J)\boldsymbol{1}
```
with the free energy being
```math
\psi(\boldsymbol{F}) = \frac{\mu}{2} ( I_1 - 3) - \mu \ln(J) +
\frac{\lambda}{2} \ln^2(J)
```
Would this approach be doable with jax-fem? Especially the automatic linearization of the current configuration integral would be of interest to me.