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

Abstract

In this paper the non-canonical Hamiltonian dynamics of a gyrostat in the three body problem will be examined. By means of geometric-mechanics methods some relative equilibria of the dynamics of a gyrostat in Newtonian interaction with two rigid bodies will be studied. Taking advantage of the results obtained in previous papers, working on the reduced problem, the bifurcations of these relative equilibria will be studied. The instability of Eulerian relative equilibria if the gyrostat is close to a sphere is proven. Necessary and sufficient conditions will be provided for lineal stability of Lagrangian relative equilibria if the gyrostat is close to a sphere. The analysis is done in vectorial form avoiding the use of canonical variables and the tedious expressions associated with them. In this way, the classic results on equilibria of the three-body problem, many of them obtained by other authors who had used of more classic techniques, are generalized.

Dynamics of a triaxial gyrostat at a Lagrangian equilibrium of a binary asteroid

“…The nonlinear stability of certain rotational equilibria, called cylindrical equilibria, is studied by means of the Energy-Casimir method (Ortega and Ratiu 1999). Examples of the application of the Energy-Casimir method to similar problems can be found in Vera (2008Vera ( , 2009). Using the tangent flow of the dynamics in the cylindrical equilibria, we obtain the necessary conditions for their nonlinear stability.…”

Stability of equilibrium states in the Zhukovski case of heavy gyrostat using algebraic methods

“…Assume that we have q 1 conserved quantities F 1 D F 1 .y/, : : : , F q D F q .y/ for (4). These functions are conserved quantities for (3). We suppose that we have also k q 1 conserved quantities F qC1 D F qC1 .y, z/, : : : , F k D F k .y, z/ for the system (3).…”

“…These functions are conserved quantities for (3). We suppose that we have also k q 1 conserved quantities F qC1 D F qC1 .y, z/, : : : , F k D F k .y, z/ for the system (3).…”

“…In Section 2, we study the stability of the equilibrium points of a differential system of the form (3). First, we present a result about the spectral stability of equilibrium points of a differential system of the form (3).…”

“…First, we present a result about the spectral stability of equilibrium points of a differential system of the form (3). Using the theory described in [6], we prove a theorem about the stability with respect to the set of conserved quantities fF 1 , : : : , F k g of an equilibrium point of a system of the form (3). An equilibrium point is stable with respect to the set of conserved quantities fF 1 , : : : , F k g if and only if we can construct, in a neighborhood of the equilibrium point, a Lyapunov function by using the conserved quantities.…”