  This paper showed that Poisson brackets in quaternion variables can be
obtained directly from canonical Poisson brackets on cotangent bundle of
$SE(3)$ (or $SO(3)$) endowed by canonical symplectic geometry. Quaternion
parameters in our case are just dynamic variables in canonical Hamiltonian
mechanics of a rigid body on $T^*SE(3)$ The obtained results based on
quaternions representation as explicit functions of rotation matrix elements of
$SO(3)$ group. The relation of obtained Poisson structure to the canonical
Poisson and symplectic structures on $T^*S^3$ were investigated. To derive the
motion equations of Hamiltonian dynamics in quaternionic variables it is
proposed to use the mixed frame of reference where translational degrees of
freedom describes in the inertial frame of reference and degree of rotational
freedom in the body frame. It turns out that motion equations for system with
Hamiltonian of the rather general form can be written in algebraic operations
on quaternions.
