Abstract
This and the following chapter compile most of the standard problems in classical mechanics. After a brief discussion of one-dimensional motion, we turn to the two-body problem.
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Notes
- 1.
Kepler, Johannes, German mathematician and astronomer, *27.12.1571 Weil der Stadt, †15.11.1630 Regensburg; he discovered the laws of planetary motion.
- 2.
This is the so called Kepler problem .
- 3.
Coulomb, Charles Augustin de, French physicist and engineer, *Angoulême 14.6.1736, †Paris 23.8.1806.
- 4.
Rutherford, Ernest, Lord Rutherford of Nelson (since 1931), British physicist, *Brightwater (Newsealand) 30.8.1871, †Cambridge 19.10.1937; probably the most influential experimental physicist of his time in the area of nuclear research. He received the Nobel Price in chemistry in 1908.
- 5.
This follows also directly from (5.10), because \({\vec {r}}_{12}={\vec {r}} (S)\) and \({\vec {R}} =0\) in the center of mass frame.
- 6.
Equation (5.98) follows via (5.96), i.e.
$$\begin{aligned} \nonumber p\,^\prime _1 \left( L \right) \cos \theta _1 = \mu v \cos \theta + \frac{\mu }{m_2} p_1 \left( L \right) \qquad \text{ and } \qquad p\,^\prime _1 \left( L \right) \sin \theta _1 = \mu v \sin \theta \;. \end{aligned}$$Next we eliminate \( p\,^\prime _1 \left( L \right) \) and make use of \(p_1 \left( L \right) = m_1 v\).
- 7.
Again we have used \(\sin \theta _2 = 2 \sin \left( \theta _2 /2 \right) \cos \left( \theta _2 /2 \right) \).
- 8.
Cassini, Giovanni Domenico, French astronomer, *Perinaldo (near Nizza) 8.6.1625, †Paris 14.9.1712.
- 9.
References
M.R. Spiegel, Advanced Mathematics - Schaum’s Outline Series in Mathematics McGraw-Hill (1971)
I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products. Academic Press (1980)
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Hentschke, R. (2017). Integrating the Equations of Motion. In: Classical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-48710-6_5
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