  A system of N particles eN=(x1,v1,...,xN,vN) interacting self-consistently
with M waves Zn=An*exp(iTn) is considered. Hamiltonian dynamics transports
initial data (eN(0),Zn(0)) to (eN(t),Zn(t)). In the limit of an infinite number
of particles, a Vlasov-like kinetic equation is generated for the distribution
function f(x,v,t), coupled to envelope equations for the M waves. Any initial
data (f(0),Z(0)) with finite energy is transported to a unique (f(t),Z(t)).
Moreover, for any time T>0, given a sequence of initial data with N particles
distributed so that the particle distribution fN(0)-->f(O) weakly and with
Zn(0)-->Z(O) as N tends to infinity, the states generated by the Hamiltonian
dynamics at all time 0<t<T are such that (eN(t),Zn(t)) converges weakly to
(f(t),Z(t)). Comments: Kinetic theory, Plasma physics.
