Abstract
In the theory of dynamical systems, topological methods are often employed when the dynamics is too complicated to answer questions like the one about the existence of periodic orbits directly.
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Notes
- 1.
in technical terms: There exists a symplectic isotopy \(\Phi _t: {\mathbb R}^{2n}\rightarrow {\mathbb R}^{2n},\ t\in [0,1]\) with \(\Phi _0=\text {Id} \) that leaves the punctured hypersurface invariant and for which \(\Phi _1(B_r)\) lies in the opposite component of \({\mathbb R}^{2n}\setminus H\) from the one containing \(B_r\).
- 2.
Its dimension is \(n(2n+3)\), because \(\dim \text {Sp} (E,\omega ) = n(2n+1)\) by Exercise 6.26.
- 3.
We take the Lebesgue measure on A, respectively the area form \(dr\wedge d\varphi \), but one may also take the measure \(r \, dr\wedge d\varphi \) that arises from polar coordinates on \({\mathbb R}^2\).
- 4.
Jordan Curve Theorem: The complement of the image \(c(S^1)\subset {\mathbb R}^2\) of a simple closed curve \(c:S^1\rightarrow {\mathbb R}^2\) in the plane consists of exactly two connected components, one of which is bounded, and the other of which is unbounded. \(c(S^1)\) is the common boundary of both connected components.
- 5.
If c is parametrized by arclength (which is possible for regular curves and will be assumed in the sequel), then positive curvature means \(\left\langle c''(t),\left( {\begin{matrix}0&{} -1\\ 1&{}0\end{matrix}}\right) c'(t)\right\rangle >0\).
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Knauf, A. (2018). Symplectic Topology. In: Mathematical Physics: Classical Mechanics. UNITEXT(), vol 109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55774-7_17
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DOI: https://doi.org/10.1007/978-3-662-55774-7_17
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