I apologize if this question is too elementary for mathoverflow; I asked it (unsuccessfully) on MATH.SE first.

As a bit of background: one way to study the mechanics of deformation of a continuous solid body is by defining a reference body $B_0$, a connected, well-behaved subset of $R^2$ or $R^3$. In 3D one assumes that the boundary $\partial B_0$ is a closed, orientable surface; in 2D it is a closed curve.

A motion is then a map $\psi : B_0 \rightarrow B$, where $\psi = \psi(\bar{X},t)$, $t > 0$ and $\bar{X}$ is some label (usually a position vector) for points in the reference body $B_0$. In particular, motions are treated as diffeomorphisms parametrized by $t$.

A particular subset of motions involve those that conserve volumes / areas. For instance, in a planar motion, the area $dv^a \times dv^b$ is conserved by the motion for all time $t$. A general incompressible motion would, if I am not mistaken, preserve a volume form in 3D.

In 2D, the area preserving motions are symplectomorphisms, not just diffeomorphisms. One expects that this imposes strong restrictions on the map $\psi(\bar{X},t)$. In the odd-dimension (3D), one expects similarly strong constraints from the conservation of volume.

The governing PDE for continuous bodies is the Cauchy equation of motion

$$ \nabla \cdot \sigma + \rho \bar{b} = \rho \left ( \frac{\partial \bar{v}}{\partial t} + \bar{v} \cdot \nabla \bar{v} \right) $$

The velocity $\bar{v}$ is defined as $\left. \dfrac{\partial \psi}{\partial t} \right|_X$ and $\bar{b}$ is a specified vector field.

In general, the stress tensor $\sigma$ at a point is a functional of the history of motion of a material point. Specifying it completes the physical description of the continuous body e.g. a fluid, elastic solid etc.

My questions are as follows:

(1) Is the presence of an incompressibility condition associated with any non-trivial, scalar, conserved quantities in the Cauchy momentum equation?

(2) It is known that there are some obvious restrictions on the velocity field, for instance $\nabla \cdot \bar{v} = 0$. However, does incompressibility also impose other, non-obvious restrictions of any kind e.g. on geometric quantities associated with the motion.

Alternatively, is there a reason that symplectomorphisms / conserved volume vs diffeomorphisms just does not yield too much more useful information as far as the Cauchy momentum equation goes?

Added: I was able to find a lot of material for a very particular case, that of incompressible fluid flow (the Euler equations), not much for the more general situation I am interested in.