The Hamiltonian formulation of continuum Calogero-Moser models
Recent well-posedness results have identified the Hardy space $L^2_+$ as the natural phase space for continuum Calogero-Moser models, both focusing and defocusing, on the line and on the torus. In this paper, we introduce a symplectic form on this phase space and so are able to realize these models as Hamiltonian systems. Moreover, we demonstrate that previously identified conserved quantities are mutually commuting, reinforcing the notion that these models are completely integrable. We further illustrate the utility of these structures by using them to give a new proof of global well-posedness in the critical space $L^2_+$, under the necessary mass restriction in the focusing case. Our work also brings to light several unforeseen connections: (i) the threshold for well-posedness coincides with that for the nondegeneracy of the symplectic form; (ii) this threshold is connected through Carleman's inequality to the isoperimetric problem in the plane; (iii) the transition from the line to the torus gives rise to a modified dynamical equation.