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arXiv · 2206.07589

A rigorous derivation of the Hamiltonian structure for the Vlasov equation

Joseph K. Miller·Andrea R. Nahmod·Nataša Pavlović·Matthew Rosenzweig·Gigliola Staffilani
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Abstract

We consider the Vlasov equation in any spatial dimension, which has long been known to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie-Poisson type. In parallel, it is classical that the Vlasov equation is a mean-field limit for a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. One may view this work as a classical counterpart to arXiv:1908.03847, which provided a rigorous derivation of the Hamiltonian structure of the cubic nonlinear Schr\"odinger equation from the many-body problem for interacting bosons in a certain infinite particle number limit, the first result of its kind. In particular, our work settles a question of Marsden, Morrison, and Weinstein on providing a "statistical basis" for the bracket structure of the Vlasov equation.

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Joseph K. Miller, Andrea R. Nahmod, Nataša Pavlović, Matthew Rosenzweig, Gigliola Staffilani. 2022-06-15. A rigorous derivation of the Hamiltonian structure for the Vlasov equation. https://arxiv.org/abs/2206.07589

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