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Hamiltonian systems, symplectic flows, classical integrable systems

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Is diffeomorphism of symplectic manifolds which preserves Lagrangian submanifolds and $J$-ho...

Consider a diffeomorphism of two symplectic manifolds $M_1$ and $M_2$ with compatible J structures, which sends Lagrangian submanifolds of $M_1$ to Lagrangian submanifolds of $M_2$. Assume moreover th …