coisotropic submanifolds

Question

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 $T^*\mathbb{R}^n$?

Answer 1

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.

Answer 2

I think that it's possible that the OP was interested in knowing something about the 'generality' of coisotropic submanifolds, even though, as José and Nicola point out, it is hard to give an explicit 'global parametrization' of them. However, if you are willing to settle for a 'local' parametrization, there is something you can say:

Remember, in the case of a Lagrangian that can locally be written as a graph of the form $\xi_k = f_k(x^1,\ldots,x^n)$, you can show that, at least locally, it follows that the $n$-functions $f_k$ can be expressed in terms of a single function, i.e., $$ f_k = \frac{\partial f}{\partial x^k} $$ where $f$ is a single function of $n$ variables.

One would like a similar description of the coisotropics of codimension $k$ that can be written locally as graphs $\xi_j = f_j(\xi_1,\ldots,\xi_{n-k},x^1,\ldots, x^n)$ for $n{-}k < j \le n$. The answer is that the $k$ functions $f_j$ for $n{-}k < j \le n$ satisfy a system of PDE that turns out to be involutive, and the general solution depends on $1$ function of $2n{-}k$ variables, $1$ function of $2n{-}k{-}1$ variables, $\ldots$, and $1$ function of $2n{-}2k{+}1$ variables, in the sense that there is a unique solution of this system when one (freely) specifies the following functions $$\begin{aligned} f_n(&\xi_1,\ldots,\xi_{n-k},x^1,\ldots\ldots\ldots\ldots, x^n),\\ f_{n-1}(&\xi_1,\ldots,\xi_{n-k},x^1,\ldots\ldots\ldots, x^{n-1},0),\\ f_{n-2}(&\xi_1,\ldots,\xi_{n-k},x^1,\ldots\ldots, x^{n-2},0,0),\\ &\qquad\qquad\vdots\\ f_{n-k+1}(&\xi_1,\ldots,\xi_{n-k},x^1,\ldots, x^{n-k+1},0,\ldots,0). \end{aligned} $$ I'm not sure how to write down the explicit solution of the PDE system (except when $k=1$ or $k=n$), but this tells you how much freedom there is in locally specifying the coisotropic submanifolds of codimension $k>0$ near the `reference' example $\xi_{n-k+1}=\xi_{n-k+2}=\cdots =\xi_{n}=0$. Moreover, near any point, a coisotropic manifold of codimension $k$ is equal to this reference example in some choice of Darboux coordinates, so this data describes the local generality of coisotropic submanifolds of codimension $k\le n$.

(Note that this works even when $k=n$, even though it gives a different description of the local solutions than the classical one above. It's an exercise for the reader to show that these two descriptions yield the same thing.)