Abstract
The two-body problem is the study of the motion of two material points \( \mathcal{P}_1 \) and \( \mathcal{P}_2 \), with masses respectively m1 and m2; when the two bodies are subject to the mutual gravitational attraction one speaks of Kepler’s problem, whose dynamics is described by the three so-called Kepler’s laws (see, e.g., [157]). In this chapter we concentrate on the mathematical description of the two-body problem. The starting point is the gravitational law and Newton’s three laws of dynamics. The gravitational law states that two bodies attract each other through a force which is directly proportional to the product of the masses and inversely proportional to the square of their distance r:
where \( \mathcal{G} \) is the gravitational constant, amounting to \( \mathcal{G} = 6.67 \cdot 10^{ - 11} m^3 kg^{ - 1} s^{ - 2} \), and \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{e} _{12} \) is the unit vector joining the two bodies. Newton’s laws of dynamics can be stated as follows:
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(i)
First law (law of inertia): without external forces every body remains at rest or moves uniformly on a straight line.
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(ii)
Second law: the net force experienced by a body is equal to the rate of change of its momentum.
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(iii)
Third law (action and reaction principle): for every action, there is an equal and opposite reaction.
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© 2010 Praxis Publishing Ltd, Chichester, UK
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Celletti, A. (2010). Kepler’s problem. In: Stability and Chaos in Celestial Mechanics. Springer Praxis Books. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85146-2_3
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DOI: https://doi.org/10.1007/978-3-540-85146-2_3
Publisher Name: Springer, Berlin, Heidelberg
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