Abstract
This Chapter is devoted to the classical results, mostly presented in the pioneer works by Mikhail Lidov and Yoshihide Kozai. The secular Lidov-Kozai Hamiltonian is considered. The classical models and analysis are described; besides, possibilities for the suppression of the Lidov-Kozai effect in various dynamical situations are discussed.
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Notes
- 1.
Nikolay Dmitrievich Moiseev (1902–1955), a professor of the Moscow University, was the founder of the Moscow school of celestial mechanics. The mentioned papers were typeset in 1941, but, due to calamities of the war, the publication was delayed until 1945.
- 2.
This follows from the so-called D’Alembert rules, specifying which combinations of angles can be present in the Fourier expansions of perturbing functions. The formulation of the D’Alembert rules is given in section 1.9.3 in Morbidelli (2002).
- 3.
The vertical line c 2 = 0 is analogous, in such a way, to the separatrix of the nonlinear mathematical pendulum: it separates the regimes of librations and circulations of an angle.
- 4.
Note that this value is the critical inclination in the considered model. If the problem is non-hierarchical (say, as in a real asteroid–Jupiter–Sun system), or the body which the particle orbits is oblate (say, as in a real satellite–planet–Sun system), the critical inclinations would be different.
- 5.
Note that when we speak here on the constancy of any element, the long-term (average) behaviour in the double-averaged problem is implied. In the single-averaged problem (and, of course, in the original non-averaged problem), the solution oscillates around the mean values given by the solution of the double-averaged problem.
- 6.
The analogous well-known separatrix of the mathematical pendulum is illustrated in Fig. 4.5
- 7.
Note that the weakly elliptic R3BP was considered already in the pioneering work by Lidov (1961).
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Shevchenko, I.I. (2017). Classical Results. In: The Lidov-Kozai Effect - Applications in Exoplanet Research and Dynamical Astronomy. Astrophysics and Space Science Library, vol 441. Springer, Cham. https://doi.org/10.1007/978-3-319-43522-0_3
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