why is a Lagrangian submanifold a semi-classical state and not a classical state?
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Let (M,ω,H)be an integrable system of dimension 2nwith integrals of motion f1=H, f2,…,fn. Let c∈Rn be a regular value of f:=(f1,…,fn). Then the corresponding level f−1(c) is a Lagrangian submanifold of M.
Geometrically this means that, locally around the regular value c, the map f:M→Rn collecting the integrals of motion is a Lagrangian fibration, i.e. it is locally trivial and the fibres are Lagrangian submanifolds.
Furthermore, one also shows that the connected components of f−1(c) are of the form Rn−k×Tk, where 0≤k≤n and Tk is a k-dimensional torus. In particular, every compact component must be a lagrangian torus.
Geometrically this means that, locally around the regular value c, the map f:M→Rn collecting the integrals of motion is a Lagrangian fibration, i.e. it is locally trivial and the fibres are Lagrangian submanifolds.
Furthermore, one also shows that the connected components of f−1(c) are of the form Rn−k×Tk, where 0≤k≤n and Tk is a k-dimensional torus. In particular, every compact component must be a lagrangian torus.
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