  We investigate the classical phase space structure of $N$ $SU(n+1)$
non-Abelian Chern-Simons (NACS) particles by first constructing the product
space of associated $SU(n+1)$ bundle with ${\bf CP}^n$ as the fiber. We
calculate the Poisson bracket using the symplectic structure on the associated
bundle and find that the minimal substitution in the presence of external gauge
fields is equivalent to the modification of symplectic structure by the
addition of field strength two form. Then, we take a direct product of the
associated bundle by the space of all connections and choose a specific
connection by the condition of vanishing momentum map corresponding to the
gauge transformation, thus recovering the quantum mechanical model of NACS
particles in Ref.\cite{lo1}.
