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In this paper we consider the ``symplectic'' version of the Euler equationsstudied by Ebin \cite{ebin}. We show that these equations are globallywell-posed on the Sobolev space $H^s(\mathbb{R}^{2n})$ for $n \geq 1$ and $s >2n/2+1$. The mechanism underlying global well-posedness has similarities to thecase of the 2D Euler equations. Moreover we consider the group ofsymplectomorphisms $\mathcal{D}^s_\omega(\mathbb{R}^{2n})$ of Sobolev type$H^s$ preserving the symplectic form $\omega=dx_1 \wedge dx_2 + \ldots +dx_{2n-1} \wedge dx_{2n}$. We show that $\mathcal{D}^s_\omega(\mathbb{R}^{2n})$is a closed analytic submanifold of the full group$\mathcal{D}^s(\mathbb{R}^{2n})$ of diffeomorphisms of Sobolev type $H^s$preserving the orientation. We prove that the symplectic version of the Eulerequations has a Lagrangian formulation on$\mathcal{D}^s_\omega(\mathbb{R}^{2n})$ as an analytic second order ODE in themanner of the Euler-Arnold formalism \cite{arnold}. In contrast to this``smooth'' behaviour in Lagrangian coordinates we show that it has a very``rough'' behaviour in Eulerian coordinates. To be precise we show that thetime $T > 0$ solution map $u_0 \mapsto u(T)$ mapping the initial value of thesolution to its time $T$ value is nowhere locally uniformly continuous. Inparticular the solution map is nowhere locally Lipschitz.

The group of Symplectomorphisms of $\mathbb{R}^{2n}$ and the Euler equations