Abstract
“And he was told to tell the truth, otherwise one would have recourse to torture. [He replied:] I am here to obey, but I have not held this opinion after the determination was made, as I said.”
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Notes
- 1.
See Sobel [Sob2, Section 24].
- 2.
In Problem 12.37 on page 308, we show that for the centrally symmetric distribution of mass, the gravitation has the same effect as if all mass were concentrated in the center.
- 3.
See Example 12.39 on page 310.
- 4.
See http://www.newtonproject.sussex.ac.uk/view/texts/normalized/NATP00180. Quotation from the introduction of Arnol’d’s book [Ar3]. Interesting biographical notes on Newton can be found in [Ar6].
- 5.
If \(\dot{x}\) is parallel to x, we obtain a straight line instead of a plane.
- 6.
To memorize which quantity to square: Times Square.
- 7.
It follows from the formulas for the semiaxes that their ratio b/a is \(1+\mathcal{O}(e^2)\), whereas the ratio \(r_{\min }/r_{\max }\) depends on the eccentricity e in the form \(\frac{1-e}{1+e} = 1-2e + \mathcal{O}(e^2)\). Planetary orbits with eccentricities \(e<{\textstyle \frac{1}{4}}\) are therefore, in first approximation, circles, where the sun is shifted by order e from the center. The ellipse in the figure has eccentricity \(e={\textstyle \frac{1}{2}}\), whereas the earth has \(e=0.017\).
- 8.
The parametrization of the conics by time is obtained by elliptic integrals.
- 9.
in units that make the constant of gravity 1.
- 10.
See Chapter 11, beginning on page 256.
- 11.
See Chapter 14, beginning on page 365.
- 12.
This will be shown in Section 6.3.3 on page 122.
- 13.
KAM theory will be discussed in Section 15.4, beginning on page 412.
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Knauf, A. (2018). Introduction. In: Mathematical Physics: Classical Mechanics. UNITEXT(), vol 109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55774-7_1
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