The Hamiltonian Dynamics of The Two Gyrostats Problem

Abstract

The problem of two gyrostats in a central force field is considered. We prove that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system posseses symmetries. Using them we perform the reduction of the number of degrees of freedom. We show that at every stage of the reduction process, equations of motion are Hamiltonian and give explicit forms corresponding to non-canonical Poissson brackets. Finally, we study the case where one of the gyrostats has null gyrostatic momentum and we study the zero and the second order approximation, showing that all equilibria are unstable in the zero order approximation.

Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria

“…An overview of the problem of two rigid bodies is found in Barkin et al (1982); and the non-canonical Hamiltonian dynamics of the two rigid bodies problem was considered by Maciejewski (1995). These papers have been generalized to the case of a gyrostat in a central Newtonian field by Wang et al (1995) and by Mondéjar and Vigueras (1999) in the case of two gyrostats in mutual Newtonian attraction.…”

Dynamics of a gyrostat on cylindrical and inclined Eulerian equilibria in the three-body problem

“…Results of [3,10] are generalized in Mondéjar et al [4] to the case of two gyrostats in mutual Newtonian attraction.…”

Eulerian Equilibria of a Gyrostat in Newtonian Interaction with Two Rigid Bodies

“…If z e = (Π e 2 , Π e 1 , Π e 0 , λ e , p e λ , µ e , p e µ ) is an Eulerian relative equilibrium, the equation (8) has, at least, a real root in which the functions h 1 (ρ) and h 2 (ρ) are given by the expressions (4,6). The modulus of the angular velocity of the gyrostat is…”

“…When | λ e | has a fixed value, let ρ be a solution of the equation (8), where the functions h 1 (ρ) and h 2 (ρ) are given by the expressions (4,6). Then z e = (Π e 2 , Π e 1 , Π e 0 , λ e , p e λ , µ e , p e µ ) given by…”

“…Let us remember that a gyrostat is a mechanical system S composed of a rigid body S ′ and other bodies S ′′ (deformable or rigid), connected to it in such a way that their relative motion with respect to its rigid part do not change the distribution of mass of the total system S (Leimanis, [2]). These papers have been generalized by Mondéjar and Vigueras [4] to the case of two gyrostats in mutual Newtonian attraction.…”