Symplectic manifolds (a problem in Arnold's classical mechanics book)

Question

I'm working my way through V.I. Arnold's Mathematical Methods of Classical Mechanics. In the process I'm trying to wrap my head around one of the problems in the chapter on Symplectic Manifolds. In particular, the problem I'm struggling with is in Chapter 8, Section 37, C (page 203 in my second edition). It reads:

In $\mathbb{R}^{2n} = {(\mathbf{p},\mathbf{q})}$ we will identify vectors and 1-forms by using the euclidean structure $(\mathbf{x},\mathbf{x}) = \mathbf{p}^2+\mathbf{q}^2$. Then the correspondence $\mathbf{\xi} \rightarrow \omega_\xi^1$ determines a transformation $\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}$. Calculate the matrix of this transformation in the basis $\mathbf{p}, \mathbf{q}$.

The solution is $ \left(\begin{matrix} 0 & E \\ -E & 0 \end{matrix}\right)$.

I think I'm lost because I don't know how to make use of the euclidean structure. How can I use it to identify vectors (or forms)? Any hints on how to approach the problem would be greatly appreciated!

Answer 1

The example immediately prior in the book gives you the symplectic structure you are meant to consider over $R^{2n}$. When he explains the "euclidean structure" on the vectors and 1-forms, you are meant to understand that the coordinate system (p,q) is a linear one. So you can expect the transformation from the vectors in $R^{2n}$ to the coordinates of the 1-form (a linear functional) to be a linear map.

Answer 2

$$\begin{align} \omega^2(\eta, \xi) &= dp\wedge dq(\eta, \xi) \\ &= \eta_p\xi_q - p_\xi q_\eta \\ &= \left\langle \begin{pmatrix} 0 & E \\ -E & 0 \end{pmatrix}\xi, \eta\right\rangle. \\ \end{align} $$