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By Rudolf Dvorak, F. Freistetter, Jürgen Kurths

This booklet is meant as an advent to the sector of planetary structures on the postgraduate point. It includes 4 broad lectures on Hamiltonian dynamics, celestial mechanics, the constitution of extrasolar planetary platforms and the formation of planets. As such, this quantity is very compatible should you have to comprehend the enormous connections among those various topics.

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This derivation of the Lagrange and Hamilton equations closely follows the very good presentation of the topic in [22]. 2 Lagrange Equations of the First Kind Although the Newtonian axioms are very important, for certain problems it is not always possible to apply them directly. If we investigate a plain pendulum with a length l, moving on a circular path, because of the restriction due to the length it follows (in rectangular coordinates) that x2 + y 2 − l2 = 0 (70) Thus, the thread induces a constraint force C and the second Newtonian axiom reads m¨ r =F +C (71) where r = (x, y, z).

For an easier treatment of the equations, we first set k 2 (M + m) = κ2 . ] will denote vectorial multiplications. Multiplying (198) with p gives ¨] = − [p, p κ2 [p, p] = 0 r3 (199) Thus we introduce the constant angular momentum vector g which is perpendicular to the plane of motion given by ˙ g = [p, p] (200) and | g |= c. The three components of this vector deliver the first three constants of motion. An additional constant can be found by [¨ p, g] = −κ2 ˙ [p, g] [p, [p, p]] = −κ2 3 3 r r (201) By solving the multiple vectorial multiplication one obtains [¨ p, g] = − κ2 ˙ p − (pp) p) ˙ ((pp) r3 (202) ˙ with rr Substituting pp ˙ and pp with r2 gives [¨ p, g] = − κ2 κ2 2 ˙ p = ( rr) ˙ p − r (rp˙ − rp) ˙ r3 r2 and thus [¨ p, g] = κ2 d dt p r (203) (204) Because of d ˙ g] = [¨ ˙ g] ˙ [p, p, g] + [p, dt with (204) one obtains by integrating ˙ g] = [p, κ2 p+f r (205) (206) The constant vector f stemming from the integration is the so-called Laplace vector, which is perpendicular to g and has | f |= d = const.

Additionally, all constraints have to be fulfilled for all values of the qi : 13 The first particle has the coordinates (x1 , x2 , x3 ), the second one (x4 , x5 , x6 ); the last one is given by (x3N −2 , x3N −1 , x3N ). 26 Rudolf Dvorak and Florian Freistetter hi (x1 (q1 , . . , qf , t) , . . , x3N (q1 , . . , qf , t) , t) = 0 (87) Using the generalized coordinates, we can now eliminate the constraint forces. The constraints in (87) do not depend on qi because, for every value of qi , hi = 0. That means that the total derivative of hi has to vanish: and thus dhi =0 dqk (88) ∂hi ∂xn =0 ∂x n ∂qk n=1 (89) 3N n with k = 1, 2, .

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