User ivah - MathOverflow

A vector field on a symplectic submanifold intersecting the symplectic complement

2012-06-26T16:08:10Z

Consider a 4-dim symplectic vector field $X$ on the symplectic manifold $(M, \omega)$ in $\mathbb{R}^4$ with $\omega= \sum_{i=1}^2 dy_i \wedge dx_i$. Moreover, the linear terms of $X$ are given by $y_1 \partial / \partial x_1 - x_1 \partial / \partial y_1$ (so not dependent on $(x_2,y_2)$). Let $X|_{\mathcal{N}}$ be the symplectic vector field restricted to

$(\mathcal{N}, \omega|_{\mathcal{N}})$ a 2-dim symplectic submanifold of $(M, \omega)$. Furthermore, suppose that $T_0\mathcal{N}$ lies in the symplectic complement of the tangent space of the plane $(x_2,y_2)$. Does there exist a symplectic transformation such that in an open neighbourhood of zero the $(\mathcal{N}, \omega|_{\mathcal{N}})$ is given by the symplectic manifold $(\hat{\mathcal{N}}, \varpi)$ in $\mathbb{R}^2$ with

\begin{equation} \varpi= d\hat{y}_1 \wedge d\hat{x}_1
\end{equation}

and such that the linear terms of $X|_{\hat{\mathcal{N}}}$ are given by
\begin{equation} \hat{y}_1 \partial / \partial \hat{x}_1 - \hat{x}_1 \partial / \partial \hat{y}_1 \end{equation}

The existence of a 2-form follows by Darboux's theorem but I don't see how to construct a symplectic transformation such that the linear terms become as above.

Any help is welcome.

Thanks in advance.

Transversal intersection of a symplectic manifold with a plane

2012-05-03T23:36:43Z

Perhaps this is stupid question.

Let $\mathcal{M}$ be a symplectic manifold in $\mathbb{R}^4$ of codimension 2 with the symplectic 2-form $dx_1 \wedge dy_1 + dx_2 \wedge dy_2$. Suppose that $\mathcal{M}$ intersects the $(x_2,y_2)$-plane perpendicular in the origin. Does there exist a symplectic transformation $(x_2,y_2) \mapsto (\hat{x}_2,\hat{y}_2) $ such that for an open neighbourhood of the origin in $\mathcal{M}$ the $ \hat{x}_2,\hat{y}_2$ are constant zero?

Or is this complete nonsense?

Symplectic submanifolds and first integrals

2012-05-03T19:47:43Z

I was working with symplectic submanifolds when I posed the following question:

Suppose I have a Hamiltonian system with the phase space $\mathcal{M}$, a symplectic manifold with the standard symplectic form. Now assume that the Hamiltonian system has two first integrals $C_1,C_2$. Define the restricted phase space $\mathcal{N}$ of $\mathcal{M}$ by taking $C_1$=constant,$C_2$=constant. What kind of conditions does $C_1$ and $C_2$ need to satisfy such that $\mathcal{N}$ is a symplectic submanifold?

Any help is welcome.