By Rudolf Dvorak, F. Freistetter, Jürgen Kurths

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**Example text**

According to (212), (198) can also be written as r c2 =− 3 p 2 r r p (215) c2 √ →c=κ p p (216) ¨ = −κ2 p with κ2 = From (158) and (148) it follows that c2 = n2 a3 p and because of (159) we get (217) 48 Rudolf Dvorak and Florian Freistetter κ2 = n2 a3 = 4π 2 3 a U2 (218) which gives the third law of Kepler: a3i κ2 = 2 Ui 4π 2 (219) where the index i stands now for the i-th body. Comparing equation (215) with (198) shows that the constant κ is in fact the quantity introduced above κ2 = k 2 (M + m) (220) but the actual value of k is still unknown and can only be determined by astronomical observations or experimental measurements.

In the following demonstration we follow closely to the very well book by [64]. The restricted three body problem is deﬁned as follows: • two bodies, named primaries, with the masses m = 0 and µ = 0 move on circular orbits, • a third, massless body m3 = 0 moves in the same plane as the primaries. The restricted three body problem can serve as a good dynamical model for the investigation of many diﬀerent types of motion in the Solar system and in other planetary systems: 52 Rudolf Dvorak and Florian Freistetter • the motion of an asteroid, with the Sun and Jupiter as primaries, • the motion of a satellite with the Sun and the planet as primaries, • the motion of a comet with the Sun and Jupiter (which is – due to its large mass – the principle perturbing body) as primaries, • the motion of terrestrial planets in extrasolar planetary systems with the star and a large Jupiter-like planet as primaries (or motions in double stars), • ...

13 These are now 3N + R equations for 3N + R unknown functions and therefore the equations can be solved. Equations (82) and (83) are called Lagrange equations of the ﬁrst kind. 3 Lagrange Equations of Second Kind The Lagrange equations of ﬁrst kind (82)–(83) were deﬁned only for cartesian coordinates. For R constraints, only f = 3N − R (84) of the 3N cartesian coordinates are independent (f is called the number of degrees of freedom). We now select f generalized coordinates: q1 , q2 , . . , qn (85) The qi have to be chosen in a way that the position of all particles is determined: (86) xn = xn (q1 , q2 , .