# Symplectic (contact) structure on $M_{n}(\mathbb{R})$

Assume that $n$ is an even number. What is a natural symplectic structure on $M_{n}(\mathbb{R})$, such that for every $1\leq k \leq n$ the manifold of $k$-rank matrices would be invariant under the flow of the Hamiltonian vector field correspond to the Hamiltonian "$H=$ Determinant"? (Or at least the manifold of $n-1$ rank matrices would be invariant under this Hamiltonian vector field)

We consider a contact analogy as follows:

Assume that $n$ is an odd number. Under what contact structures on $M_{n}(\mathbb{R})$, the set of singular matrices is invariant under the flow of corresponding Reeb vector field? Or a refined version: the set of $k$ rank matrices would be invariant under this flow.

For $n-1$, doesn't this follow from conservation of energy?
For $k\leq n-2$, the derivative of the Hamiltonian vanishes, so the flow is zero, so it's trivially invariant.
• @AliTaghavi I don't know anything about the literature, and I also don't know a canonical symplectic structure on $M_n(\mathbb R)$ to evaluate the dynamics in. – Will Sawin Mar 4 '15 at 23:15