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.
Joseph Lagrange (1736–1813), a French mathematician and physicist.
- 2.
William Hamilton (1805–1865), Irish mathematician, physicist, and astronomer.
- 3.
By multiplying a vector field by a suitable smooth positive function, completeness of the flow can be attained without changing the orbits. The property of being Hamiltonian will, however, not be preserved in general.
- 4.
In the book [Art] by E. Artin, there is a discussion of the symplectic algebra over arbitrary fields rather than the field \({\mathbb R}\) of real numbers.
- 5.
This is the monic polynomial \(p\in {\mathbb C}[x]\) of smallest degree for which \(p(X)=0\).
- 6.
\(A\in \mathrm{Mat}(n,{\mathbb C})\) is called normal , if \(A^\top A = AA^\top \).
- 7.
This follows, for instance, from the observation that \(S^3\) is simply connected, but \(S^1\), and hence also \(S^2\times S^1\), is not, see Definition A.22.
- 8.
For suitable orientations of \({\mathbb T}^2\) and \(S^2\) and an arbitrary regular value \(s\in S^2\) of the Gauss mapping G, the linking number equals the mapping degree
$$\begin{aligned} \deg (G) = \sum _{t\in G^{-1}(s)} {\text {sign}}\bigl (\det (\mathrm {D}G(t))\bigr ). \end{aligned}$$.
- 9.
The name is owed to the fact that, when the atoms of the two sublattices have opposite charges, an oscillation of the optical branch can generate a light wave.
- 10.
The linear vector field \(A:{\mathbb R}^d\rightarrow {\mathbb R}^d\text{, } q\mapsto Bq\) is also called vector potential of B.
The differential one form \(\sum _{k=1}^d A_k(q)dq_k=\sum _{k,\ell =1}^d B_{k,\ell }\, q_\ell \, dq_k\) associated with A has exterior derivative \(\sum _{k,\ell =1}^d B_{k,\ell }dq_\ell \wedge dq_k\). This two-form is called’magnetic field strength’. For more on this, including non-constant magnetic fields and the time dependent case, see Example B.21.
- 11.
Analogous definitions apply to \({\mathbb C}\)-vector spaces.
- 12.
Strictly speaking, \(\Phi \) becomes a coordinate map only when we identify \({\text {Lin}}(u, u_s)\) with \(\mathrm{Mat}\bigl (n\times (m-n),{\mathbb R}\bigr )\cong {\mathbb R}^{n(m-n)}\) by choosing a basis. Moreover, we use in (6.5.1) that \({\mathbb R}^n= u\oplus u_s\).
- 13.
If both the points on \(S^1\) and the subspaces \({\mathbb R}\subset {\mathbb R}^2\) are parametrized by an angle \(\varphi \) as in Example 6.52, this identity map takes the form \(\text {MA}_1:\Lambda (1)\rightarrow S^1 \text{, } \varphi \mapsto 2\varphi \) !
- 14.
See also Bott [Bo1].
- 15.
We consider the circle as \(S^1 = \{z\in {\mathbb C}\mid |z|=1\}\) and extend the natural logarithm in such a way that the mapping \(S^1\rightarrow i\,{\mathbb R} \text{, } z\mapsto \frac{\mathrm {d}}{\mathrm {d}z}\log \bigl (f(z)\bigr )\) is continuous.
- 16.
Definition: A set M is called a homogenous space , if some group G operates transitively on M. —In the present example, the Lie group \({\text {U}}(m)\) operates continuously on the set \({\text {U}}(m)/{\text {O}}(m)\) of equivalence classes of unitary matrices if we use the quotient topology coming from \({\text {U}}(m)\) (see page 484).
- 17.
Any other point of \(\Lambda (m)\) could be chosen just as well.
- 18.
In Hirsch [Hirs], one can find an extensive exposition on the subject of ‘transversality’.
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Knauf, A. (2018). Hamiltonian Equations and Symplectic Group. In: Mathematical Physics: Classical Mechanics. UNITEXT(), vol 109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55774-7_6
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