By Rudolf Dvorak, F. Freistetter, Jürgen Kurths
This e-book is meant as an creation to the sphere of planetary structures on the postgraduate point. It contains 4 huge lectures on Hamiltonian dynamics, celestial mechanics, the constitution of extrasolar planetary platforms and the formation of planets. As such, this quantity is very compatible if you happen to have to comprehend the sizeable connections among those assorted topics.
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Extra resources for Chaos and Stability in Planetary Systems
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, . . , f . Multiplying (82) with ∂x ∂qk and building the summation over n gives then 3N 3N ∂xn ∂xn mn x ¨n = Fn (90) ∂q ∂qk k n=1 n=1 with n = 1, 2, . . 14 Equation (90) now does not depend on the constraints or constraint forces.
X3N ) (91) q = (q1 , q2 , . . , qf ) x˙ = (x˙ 1 , x˙ 2 , . . , x˙ 3N ) (92) (93) q˙ = (q˙1 , q˙2 , . . , q˙f ) (94) Diﬀerentiation of (86) with respect to the time gives x˙ n = d xn (q, t) = dt f k=1 ∂xn (q, t) ∂xn (q, t) = x˙ n (q, q, q˙k + ˙ t) ∂qk ∂t (95) Thus it follows for x˙ n ∂ x˙ n (q, q, ˙ t) ∂xn (q, t) = ∂ q˙k ∂qk (96) The kinetic energy (in cartesian coordinates) can be written as 3N T = T (x) ˙ = mn 2 x˙ 2 n n=1 (97) If xn (q, t) does not explicitly depend on the time, equation (95) reduces to f x˙ n = k=1 14 ∂xn (q) q˙k ∂qk Note that xn stands for xn (q1 , q2 , .
Because [g, f ] = 0, this gives 2 more constants of motion. Now we can write Stability and Chaos in Planetary Systems ˙ g]) = (p, [p, κ (pp) + (pf ) r 47 (207) Because of ˙ g]) = (g [p, ˙ p]) = (gg) = c2 (p [p, (208) and (pf ) = rd cos(φ − φ0 ) it follows that c2 = r κ2 + d cos(φ − φ0 ) (209) c2 (κ2 + d cos(φ − φ0 )) (210) and r= This can also be written as r= c2 κ2 1+ d κ2 cos(φ − φ0 ) (211) which is identical to the ﬁrst law of Kepler (147) with p= c2 κ2 e= d κ2 (212) Substituting into the second law of Kepler (r2 dφ = cdt) gives p2 2 dφ (1 + e cos (φ − φ0 )) = cdt (213) and, after integrating φ p2 φ0 dφ 2 (1 + e cos (φ − φ0 )) = c (t − t0 ) (214) t0 , the perihelion time (the moment, when planet is in its perihelion), is now the sixth and last constant of motion.
Chaos and Stability in Planetary Systems by Rudolf Dvorak, F. Freistetter, Jürgen Kurths