  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.
