On Orbital Stability of Periodic Motions in Problems of Satellite Dynamics Bardin B.S. Moscow Aviation Institute Dept. of Theoretical Mechanics Universidad Publica de Navarra, Navarra, 2010

Orbital Stability M(t0)M(t0) M(t1)M(t1) P(t0)P(t0) P(t1)P(t1) P(t0)P(t0) M(t0)M(t0) M(t1)M(t1) P(t1)P(t1)

M(t 0 )=M(t 0 +T) P(t0)P(t0) P(t 0 +T)

Definition of orbital stability Periodic orbit Neighborhood of periodic orbit Def. A periodic orbit  is orbitally stable iff for all  >0 there exist  >0 such that if y ( t 0 ) belong to N ( ,  ) then y ( t ) belong to N ( ,  ) for all t > t 0.

Orbital Stability  

Orbital Stability of Hamiltonian Systems Autonomous Hamiltonian system Periodic solution Transformation: Periodic solution in the variables  i  i

Isoenergetic reduction The Hamiltonian in the variables  i  i is periodic in  1 Isoenergetic reduction on the level H=0: Whittaker Equations Stability of the solution  j  j =0, (j=2,…,n)

Stability of the reduced system Hamiltonian of the reduced system Linear system Normalization of K Stability analyses

Orbital stability of planar periodic motions of a satellite We consider motions of a satellite about its center of mass. Satellite is a rigid body which moments of inertia satisfy С=А+В (plate) The mass center moves in a circular orbit Unperturbed motion: axes of the minimal moment of inertia is perpendicular to the orbital plane. Two types of planar periodic motions: oscillations and rotations

Reference frames OXYZ – orbital frame: axes OX, OY, OZ are directed along the radius vector of the mass center, the normal and transversal of the orbit Oxyz – principal axes frame, fixed in satellite. Axes directed along the ellipsoid inertia axes. Euler angels  define orientation of the satellite

Hamiltonian of the problem  A = A/B,  C = C/B; From С=A+B =>  A =  C -1 Equation of motion: Hamiltonian:

Unperturbed motion. Action-angle variables Planar motions (unperturbed motion):  p  =p  =0 Unperturbed motion in the action-angle variables Equations for the unperturbed motion: Action-angle variables: General solution (oscillation):

Perturbed motion, Isoenergetic reduction Let us introduce perturbations : Hamiltonian of the perturbed motion: Isoenergetic reduction (level H=0): New independent variable: w =    I 0 

Hamiltonian of the reduced system

Linear system Matrix of fundamental solutions Initial conditions: X(0)=E ( k =1,2; j =1,2,3,4).

Study of stability of the linear system Characteristic equation Conditions of stability in linear approximation Characteristic exponents Linear normalization

Nonlinear study of stability Nonresonant case Hamiltonian normal form Criteria of stability for most of initial conditions Criteria of formal stability: Form m ( r 1, r 2 ) is definite for all Nonlinear normalization: where

Nonlinear study of stability Resonant case Hamiltonian normal form Criteria of stability in third approximation Resonant cases: