The particle motion at a presence of a large magnetic field directed along the particle trajectory demands the special description. This article deals with the decomposition of the Hamiltonian on the two parts: fast and slow motion. The first part describes the fast rotation around the magnetic line of longitudinal field. The second part describes the slow drift of rotation center from one magnetic line to another. The supposed method enables to write the simple Hamiltonian to each motion type and to formulate the matrix formalism for any element of an accelerator device (quadruple, skew‐ quadruple, drift gap, bend with a filed index). The Hamiltonian decomposition has physical clearness when the longitudinal field is larger than another fields but it is correct for the arbitrary parameters. At the small longitudinal field the coupling term in Hamiltonian between two modes is essential. The dispersion property of fast and slow modes is derived easy from Hamiltonian also. This method expands easily for nonlinear motion of such modes. This results may be used at analyzed the electron motion in the cooling device, the muon motion in the muon ionization cooler or another system with strong solenoidal coupling.

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