By Professor Alessandra Celletti (auth.)
From the reviews:
“The target of this publication is to illustrate a contemporary points of celestial mechanics. … The publication comprise a suite of Appendices (A–G) strong for speedy connection with the literature … . the current very good booklet is dedicated to boost point undergraduate scholars in addition to postgraduate scholars and researchers. the writer supplied accomplished literature, one hundred seventy five items.” (Nikolay Asenov Kostov, Google Bucher, June, 2010)
“This e-book is a accomplished creation to difficulties of celestial mechanics applicable for college kids, PhD scholars and experts within the area.” (Nikolay Asenov Kostov, magazine of Geometry and Symmetry in Physics, Vol. 20, 2010)
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Additional resources for Stability and Chaos in Celestial Mechanics
Sample text
1). Let F 1 be the force exerted by P2 on P1 and let F 2 be the force exerted by P1 on P2 . P1 F1 F2 B P2 R r1 r2 O Fig. 1. Distance vectors in an inertial reference frame with origin in O. By the action and reaction principle one has F 1 = −F 2 , where F1 = G m1 m2 r . 1), the equations of motion are given by the expressions m1 m2 r d2 r 1 = G 2 dt r2 r 2 d r m1 m2 r . 2) d2 r 1 d2 r 2 + m =0, 2 dt2 dt2 whose integration provides the relations: m1 dr1 dr + m2 2 = C 1 , m1 r1 + m2 r2 = C 1 t + C 2 , dt dt with C 1 , C 2 being constant vectors.
Dt Through the angular momentum integral one gets dr ϑ − ϑ0 = h r2 2(E − Ve (r)) = arccos r0 r −1 1− , E E0 where r0 is such that Ve (r0 ) is minimum and E0 = E(r0 ), namely r0 = h2 , μ E0 = − μ2 . 11) we find p = r0 , e= 1− E , E0 ϑ0 = g0 . In summary we obtain that the parameter e is related to the energy E by e= 1+ 2h2 E . e. e. e. e. e > 1): the trajectory is a hyperbola. 8 Spacecraft transfers As a practical implementation of Keplerian orbits, we consider the problem of spacecraft transfers.
4). 8). 1 Poincar´ e map An effective tool for investigating the dynamical properties of a mechanical system is obtained by introducing the so–called Poincar´e map [8], which reduces the study of a continuous system to that of a discrete mapping. 1) where f = f (z) is a generic regular vector field. Let Φ(t; z 0 ) be the flow at time t with initial condition z 0 and let Σ be an (n−1)–dimensional hypersurface transverse to the flow1 , which we shall refer to as the Poincar´e section. 1), let z p be the intersection of the periodic orbit with Σ and let U be a neighborhood of z p on Σ.
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