An algebraic method of obtaining of symplectic coordinates in a rigid body dynamics

An algebraic procedure of getting of canonical variables in a rigid body dynamics is presented. The method is based on using a structure of an algebra of Lie—Poisson brackets with which a Hamiltonian dynamics is set. In a particular case of a problem of a top in a homogeneous gravitation field the method leads to well–known Andoyer—Deprit variables. Earlier, the method of getting of them was based on geometric approach (Arhangelski, 1977). Introduction Poisson brackets play a key role in Hamiltonian mechanics. According to the theorem of Darboux (Olver, 1986), degenerated Poisson manifolds are stratified to symplectic manifolds (leaves). A reduction of a Hamiltonian system on common level of all Casimir functions leads to usual Hamiltonian mechanics. The reduction is especially algebraic problem and can be made without referring to a concrete physical problem. In the present paper an algebraic method of finding of symplectic coordinates of Lie—Poisson brackets in some important for physical application cases is demonstrated. 1 A symplectic basis of the Lie—Poisson brackets with an algebra e(3) The equations of motion of a rigid body with a fixed point in a homogeneous gravitation field are traditionally written in variables M,γ, where M is a vector of an angular momentum of the body and γ is unit vector parallel to the vector of gravitational field strength. The coordinate system is toughly connected with the body. The equations of motion are Hamiltonian with a degenerated Poisson brackets (Olver, 1986) {Mi,Mj} = −ǫijkMk, {γi, γj} = 0, {Mi, γj} = −ǫijkγk (1.1)