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Extra resources for The General and Restricted Problems of Three Bodies: Course Held at the Department of General Mechanics September 1973

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Y. , ( 1970). S. Harrington, Astron. , ZJ, 190, (1968). J. , Berlin, (1866). O. , ~' 252, (1911). R. , Milano, (1919). A. Merman, Astron. , 30, 332, (1953). A. Merman, Bull. Inst. Theor. , 2, 69, (1955). H. , New Jersey, (1966). H. Pollard, J. Z, 601, (1967). H. Pollard and D. Saari, Arch. Rational Mech. , 30, 263, (1968). H. and D. Saari, Celestial Mechanics, 1, 347, (1970). D. Saari, Trans. Amer. Math. , 162, 267, (1971). L. Siegel, Ann. , (2), ~' 127, (1941). L. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer, Berlin, (1971).

Now the three equations of motion are ready to be transformed. From the equations of motion we have .. m3r, = G 3mz -rz, + G 3r31 r31 r,z ~ rz = G m, m3 3r3z+ G -3 r,z r,z r 23 ' 36 Jacobian Coordinates Subtracting the first equation from the secondwe have u. r -r =-Gp-r3 + G m3 ( r32 r3, ) , -3- - - 3 r 31 r32 or . -;. + G ( r32 - r m3 )J 3 A substitution into the third equation of motion gives or M G p- =-)J The last equations for r and for p form a 12-th order system using Jacobian coordinates.

Consider the simplest form of Sundman 1 s inequality 2 •• c s (I - 2h) I . L + 2h s I I or 2 _c_- I s 2lhl i I --' 2lhl 45 Sundman 1 s Modified Inequality for h > 0. The sign of j is controlled by the value of _I and in troducing the critical value of the moment of inertia, 2 2~ h I =lc ' we have lc- I s So as long as I> lc, I If, on the other hand, I 2lhl I is positive. I< lc, I is larger than a negative num- ber, consequently nothing may be said regarding its sign. Considering the complete inequality, we have consequently as long as we have_l> 0.

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