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Let's say that $x:=(p,q)$ is a trajectory in phase space and $$x'(t) = J \nabla H(x(t))$$ are Hamilton's equation of motion.

Now I transform $F: M \rightarrow N, x \mapsto y(x)$ diffeomorphic to some new coordinates such that $$DF\ J \ DF^T = J$$ in what way do I define my new Hamiltonian function $K(y(x))$ such that $$y'(x(t)) = J \nabla K(y(x(t)))$$ holds?

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The hamiltonian is a function on the phase space so it transforms as such. I.e. $$K(y(x))=H(x)$$ You can derive the condition on the transformation of J precisely from the condition $$x'(t) = J \nabla_x H(x(t)) \to y'(t) = J \nabla_{y(x)} K(y(x))$$

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