  In this study, we propose a universal scenario explaining the $1/f$
fluctuation, including pink noises, in Hamiltonian dynamical systems with many
degrees of freedom under long-range interaction. In the thermodynamic limit,
the dynamics of such systems can be described by the Vlasov equation, which has
an infinite number of Casimir invariants. In a finite system, they become
pseudoinvariants, which yield quasistationary states. The dynamics then exhibit
slow motion over them, up to the timescale where the pseudo-Casimir-invariants
are effective. Such long-time correlation leads to $1/f$ fluctuations of
collective variables, as is confirmed by direct numerical simulations. The
universality of this collective $1/f$ fluctuation is demonstrated by taking a
variety of Hamiltonians and changing the range of interaction and number of
particles.
