Abstract
The major part of our knowledge about molecules, atoms, and elementary particles is obtained by scattering experiments, in which particles of a specific initial velocity collide with each other or with a fixed target.
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Notes
- 1.
Image: https://commons.wikimedia.org/wiki/File:Billard.JPG, August 2006, photo by Noé Lecocq in collaboration with H. Caps. Courtesy of Noé Lecocq.
- 2.
Quoted after Nicholas Copernicus: On the Revolutions, edited by Jerzy Dobrzycki, translation and commentary by Edward Rosen, 1978, The Johns Hopkins University Press; p 17.
- 3.
This Hamiltonian occurs in the proof of non-integrability for the classical Yang-Mills equation with gauge group \(\text {SU} (2)\).
- 4.
Christian Møller, Danish physicist (1904–1980). See his article:
General properties of the characteristic matrix in the theory of elementary particles. Danske Vid. Selsk. Mat.-Fys. Medd. 23, (1945) 1–48.
- 5.
The symmetric difference of two subsets \(A, B\subset M\) is \(A\mathbin \triangle B:=(A\setminus B)\cup (B\setminus A)\).
- 6.
If E is a regular value of H, one can show (analogous to Theorem 12.15) that the complement sets \(TS^{d-1}\setminus A_E^\pm \) have measure zero, in reference to the natural measure on \(TS^{d-1}\). These sets belong to the trapped orbits.
- 7.
Strictly speaking, the Kepler motion had to be regularized for these collision orbits. Then the expression for the Rutherford cross section is obtained also for \(\Delta \theta =\pi \).
- 8.
The Austrian mathematician Johann Radon (1887–1956) investigated this integral tranform named after him in a 1917 paper.
- 9.
Definition: The Schwartz space \(\mathcal{S}\bigl ({\mathbb R}^d\bigr )\) is the function space
$$ \mathcal{S}\bigl ({\mathbb R}^d\bigr ) := \big \{f\in C^\infty ({\mathbb R}^d,{\mathbb C}) \mid \forall \ \alpha ,\beta \in {\mathbb N}_0^d:t \mapsto t^\alpha \partial ^\beta f(t)\ \text{ is } \text{ bounded }\big \}. $$ - 10.
See also the survey article [CN] on classical and quantum mechanical time delays.
- 11.
Up to an error depending on E, which can be found in the theorems in [Nat], Chapter IV.2.
- 12.
A more realistic study of the mechanics of billiard can be found in § 27 of Sommerfeld [Som]. It includes the study of top and bottom spin, follow shots etc.
- 13.
More precisely: \(H\circ \Psi = H_N\circ \pi _1 + H_r\circ \pi _2\) with \((\pi _1,\pi _2):T^*{\mathbb R}^{2d}\rightarrow T^*{\mathbb R}^{d}_{q_N}\times T^*{\mathbb R}^{d}_{q_r}\).
- 14.
After: Henri Poincaré: New Methods of Celestial Mechanics, Daniel L. Goroff, Ed., American Institute of Physics. Page 19.
- 15.
Theorem (Sard), see Hirsch [Hirs, Chapter 3.1]: For \(f\in C^k(U, \mathbb {R}^m)\) with \(k \ge {\text {max} }(n-m+1, 1)\) and \(U\subseteq \mathbb {R}^n\) open, let \(\mathrm {Crit}(f) := \{x \in U\mid \mathrm {rank}(\mathrm {D}f(x))< m\}\) be the critical set of f. Then the set \(f\bigl (\mathrm {Crit}(f)\bigr )\subseteq \mathbb {R}^m\) of critical values has Lebesgue measure 0..
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Knauf, A. (2018). Scattering Theory. In: Mathematical Physics: Classical Mechanics. UNITEXT(), vol 109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55774-7_12
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