I'm thinking about coisotropic/involutive submanifolds of the symplectic phase space $T^\*\mathbb{R}^n$ (with coordinates $x\_1,\ldots,x\_n,\xi\_1,\ldots,\xi\_n$). As I understand, the smallest coisotropic submanifold is this setting is the Lagrangian $\{\xi\_1=\ldots=\xi\_n=0\}$ (aka the zero section). Of course, the largest coisotropic is the whole phase space. In between, a "model" coisotropic of codimension $m$ is given by the equations $\xi\_1=\ldots=\xi\_m=0$ ($\xi\_{m+1},\ldots,\xi\_n$ being allowed to vary).


In terms of $x\_1,\ldots,x\_n,\xi\_1,\ldots,\xi\_n$, what equations could I use to describe an arbitrary coisotropic submanifold of $T^\*\mathbb{R}^n$?



coisotropic submanifolds
Locally, any codimension-$k$ submanifold can be described as the zero locus of $k$ smooth functions $f\_1,\dots,f\_k$. (This is true globally if and only if the submanifold has trivial normal bundle.) A submanifold $M \subset T^\*\mathbb{R}^n$, which is cut out by $f\_1,\dots,f\_k$ is coisotropic if the functions are in involution with respect to the Poisson structure, so that the restriction of $\{f\_i,f\_j\}$ to $M$ vanishes, for all $i,j$. Necessarily $k\leq n$ in this case with the minimially coisotropic ($k=n$) corresponding to the lagrangian submanifolds. You can describe lagrangian submanifolds as the image of closed 1-forms, thought of as sections $\mathbb{R}^n \to T^\*\mathbb{R}^n$.


Based on this, I would say that probably writing general equations for coisotropic submanifolds is not going to be possible, at least with any level of concreteness.


