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Mathematical Physics: Classical Mechanics pp 11–29Cite as

Dynamical Systems

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Part of the book series: UNITEXT ((UNITEXTMAT,volume 109))

Abstract

Dynamics can be viewed under different aspects, and with a variety of additional structures; accordingly, there are different definitions of dynamical systems.

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Notes

  1. 1.

    The photos are from Chapter 10 (written by D. Fowler and P. Prusinkiewicz [FP]) in H. Meinhardt: The Algorithmic Beauty of Sea Shells. 4. ed., c Springer 2009. Photos: courtesy of D. Fowler and P. Prusinkiewicz

  2. 2.

    Translation: “We therefore must consider the present state of the universe as the consequence of its earlier and the cause of its future states. An intelligence that would know, for some given moment, all forces that animate nature, as well as the respective positions of the elements of which it consists, and which would be comprehensive enough to subject these quantities to analysis, would in one and the same formula comprise the movements of the largest celestial bodies and the lightest of atoms; nothing would be uncertain to that intelligence, and future and past would be visible to it.”

  3. 3.

    Here, \(({\mathbb R},+)\) or \(({\mathbb Z},+)\) respectively are considered as a topological group (Definition E.16), and \(G\times M\) carries the product topology (Appendix A.1).

  4. 4.

    A mathematical variation of the game of pool.

  5. 5.

    But it is only when \(\dim (\mathcal{H})<\infty \) that the phase space will be a (finite-dimensional) manifold according to the definition A.25.

  6. 6.

    See Appendix E.1 and E.2.

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Authors and Affiliations

  1. Department of Mathematics, Friedrich-Alexander University Erlangen-Nürnberg, Erlangen, Germany

    Andreas Knauf

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  1. Andreas Knauf
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Correspondence to Andreas Knauf .

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Knauf, A. (2018). Dynamical Systems. In: Mathematical Physics: Classical Mechanics. UNITEXT(), vol 109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55774-7_2

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