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In $\mathbb{R}^{2n}$, $\omega=\sum dx_i \wedge dy_i$ is a canonical symplectic form, and H is an hamiltonian function, i.e. $\dot{x}= \frac{\partial H}{\partial y}$, $\dot{y}= -\frac{\partial H}{\partial x}$ ($\star$). If we change the variables with keeping the symplectic form, i.e. $x_i=f_i(u,v),y_i=g_i(u,v)$, and $\omega = \sum df_i(u,v) \wedge dg_i(u,v) = C\sum du_i \wedge dv_i$, where $C$ is a constant. So does the equation ($\star$) still hold for (u,v)?

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The equation is just $\omega(X,Y)+\mathrm{d}H(Y)=0$ for any vector field $Y$ (just be careful with sign conventions), where one solves for $X$.

If a diffeomorphism $\phi$ is given, now it is easy to apply it on both sides of the equation and see how $X$ is written after the change of variables.

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