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By Victor Szebehely (auth.), K. B. Bhatnagar (eds.)

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Extra info for Space Dynamics and Celestial Mechanics: Proceedings of the International Workshop, Delhi, India, 14–16 November 1985

Example text

Indeed the mean anomaly £ is now ignorable in the transformed Hamiltonian. Hence on each manifold L = constant, the averaged perturbed Keplerian problem may be replaced by a system with only two degrees of freedom defined by the equations dg dt dK dh dG' dr dK dH' dG dT dK - dg , dH dT dK - dh derived from the Hamiltonian the independent variable being the long time scale T defined by the 1form dT = J 2 dt. It is said that the equations determine the osculating ellipse. The pos1t10n on the osculating ellipse is determined by effecting the quadrature d£ dT dF dL Thanks to Meyer's theorem, the conventional reduction by ignoration of a cyclic variable acquires a geometric meaning which reveals the structure of the phase space on which the reduced system is supposed to act.

S2+3 )(1-a). In the new variables, the equations of motion are of form d Xi ~ oK = oY. dK =- 1 where K = (9) ~,( i=1,2,3). (S +3) (I-a) , 1 = - 1 2 [Y2+1 22 _ . ( -y+a - - X Y + X3Y2) 2 J+ 1 -(Y +Y ) 1 X2 4 4 2 3 2aM 2 3 1 I'M o(1-a)y_ 4 X M 1 (10) Here the Lagrange solutions (7) reduce to the form Xo = 1 5. 1 ' Y~ 0, X~ = 1/2, Y~ = ° (11 ) CHARACTERISTIC ROOTS FOR THE VARIATIONAL EQUATIONSIN THE NEIGHBOURHOOD OF LAGRANGE'S SOLUTIONS Let us determine the variational equations in the neighbourhood of the equilibrium solution (11) usingothe coordinates Qi and the momenta Pi defined by the formulas X.

R. prepared the launching of manmade satellites have astronomers turned their attention to the problem of predicting the motion of a mass point at low altitudes in the field of an oblate planet. As soon as they did, though, they were confronted with puzzling features: Ci) at the critical inclinations I mod 'IT =(1) tan- 1 2, the mean motion of the average argument of perige~ vanishes ; Cii) the families of periodic orbits for the principal part (32= 0) cannot be continued analytically by the Lindstedt-Poincare method into periodic orbits of the main problem (J 2 f 0) beyond the critical incli.

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