By Rudolf Dvorak, F. Freistetter, Jürgen Kurths
This e-book is meant as an creation to the sector of planetary structures on the postgraduate point. It comprises 4 vast lectures on Hamiltonian dynamics, celestial mechanics, the constitution of extrasolar planetary platforms and the formation of planets. As such, this quantity is very appropriate in case you have to comprehend the monstrous connections among those various topics.
Read or Download Chaos and Stability in Planetary Systems PDF
Similar astrophysics & space science books
S Chandrasekhar, popularly referred to as Chandra, was once one of many premier scientists of the 20 th century. The yr 2010 marks the start centenary of Chandra. His distinct type of examine, inward sure, looking a private point of view to grasp a specific box, after which move directly to one other was once so designated that it'll draw huge curiosity and a focus between students.
The detection of radial and non-radial solar-like oscillations in hundreds of thousands of G-K giants with CoRoT and Kepler is paving the line for targeted experiences of stellar populations within the Galaxy. The on hand commonplace seismic constraints permit mostly model-independent choice of stellar radii and much, and will be used to figure out the location and age of millions of stars in several areas of the Milky method, and of giants belonging to open clusters.
- An Introduction to Astrophysical Hydrodynamics
- The Joy of Fourier
- An Assessment of Precision Time and Time Interval Science and Technology
- Astronomy in Depth
Extra info for Chaos and Stability in Planetary Systems
Xn (q, t)). Because ∂U ∂ q˙k (105) = 0, (103) reads d ∂ (T − U ) ∂ (T − U ) = dt ∂ q˙k ∂qk (106) We now deﬁne the Lagrange Function L L(q, q, ˙ t) = T (q, q, ˙ t) − U (q, t) 15 Here we use that d ∂xn dt ∂qk = ∂ x˙ n ∂qk which can be shown easily. (107) 28 Rudolf Dvorak and Florian Freistetter which is the diﬀerence between the kinetic and the potential energy. The ﬁnal form of the Lagrange equation of second kind is now d ∂L(q, q, ˙ t) ∂L(q, q, ˙ t) − =0 dt ∂ q˙k ∂qk (108) with k = 1, 2, . .
From equation (109) we get (110) q˙k = q˙k (q, p, t) We now deﬁne the Hamilton function16 : f q˙i (q, p, t) pi − L (q, q˙ (q, p, t) , t) H(q, p, t) = (111) i=1 We can use the Lagrange equation (108) to obtain the partial derivatives of H: ∂H = ∂qk f i=1 f ∂ q˙i ∂L ∂ q˙i ∂L pi − − ∂qk ∂qk i=1 ∂ q˙i ∂qk ∂L ∂qk d ∂L =− dt ∂ q˙k = −p˙k =− 16 (112) Note that the Hamilton function is obtained from the Lagrange function by a Legendre transformation. Stability and Chaos in Planetary Systems ∂H = ∂pk f i=1 f ∂ q˙i ∂L ∂ q˙i pi + q˙k − ∂pk ∂ q˙i ∂pk i=1 = q˙k ∂H = ∂t f (113) f i=1 =− 29 ∂L ∂ q˙i ∂L ∂ q˙i pi − − ∂t ∂ q˙i ∂t ∂t i=1 ∂L ∂t (114) Equations (112) and (113) are the canonical or Hamiltonian equations: ∂H(q, p, t) ∂qk ∂H(q, p, t) q˙k = ∂pk p˙k = − (115) (116) These equations follow directly from the Lagrangian equations and thus they are equivalent to the equations of motion given by (108).
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 , .