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The Hamiltonian theory for the collective longitudinally polarized gluonexcitations (plasmons) coupling with classical high-energy test color-chargedparticle propagating through a high-temperature gluon plasma is developed. Ageneralization of the Lie-Poisson bracket to the case of a continuous mediuminvolving bosonic normal field variable$a^{\phantom{\ast}\!\!a}_{\hspace{0.03cm}{\bf k}}$ and a non-Abelian colorcharge $Q^{\hspace{0.03cm}a}$ is performed and the corresponding Hamiltonequations are presented. The canonical transformations including simultaneouslyboth bosonic degrees of freedom of the soft collective excitations and degreeof freedom of hard test particle connecting with its color charge in the hotgluon plasma are written out. A complete system of the canonicity conditionsfor these transformations is derived. The notion of the plasmon number density${\mathcal N}^{a\hspace{0.03cm}a^{\prime}_{\phantom{1}}\!}_{{\bf k}}$, which isa nontrivial matrix in the color space, is introduced. An explicit form of theeffective fourth-order Hamiltonian describing elastic scattering of plasmon offa hard color particle is found and the self-consistent system of Boltzmann typekinetic equations taking into account the time evolution of the mean value ofthe color charge of the hard particle is obtained. On the basis of theseequations, a model problem of interaction of two infinitly narrow wave packetsis considered. A system of nonlinear first-order ordinary differentialequations defining the dynamics of the interaction of the colorless $N^{l}_{\bfk}$ and color $W^{l}_{\bf k}$ components of the plasmon number density isderived. The problem of determining the third- and fourth-order coefficientfunctions entering into the canonical transformations of the original bosonicvariable $a^{\phantom{\ast}\!\!a}_{{\bf k}}$ and color charge$Q^{\hspace{0.03cm}a}$ is discussed.

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: plasmon -- hard particle scattering