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Classical Hamiltonian Chaos

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Quantum Signatures of Chaos

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

This chapter will present classical Hamiltonian mechanics to the extent needed for the semiclassical endeavors to follow.

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Notes

  1. 1.

    Here and below, it is best to read \(X={q\choose p},\,\partial _X={\partial _q\choose \partial _p}\), and \(\widetilde {\partial _X}=(\partial _q,\partial _p)\).

  2. 2.

    The time dependent Hamiltonian for the linearized flow is obtained from the original H by expanding around the chosen trajectory and dropping terms of higher than second order in ΔX.

  3. 3.

    The notion of ensembles is of particular importance for the statistical treatment of many-particle systems.

  4. 4.

    The symplectic area element can also be written as the antisymmetric wedge product δX ∧ δX′ .

  5. 5.

    For the Hamiltonian dynamics under exclusive consideration here, no trajectory can end in, start out from, or pass through a stationary point such that neither F(X 0) = 0 nor F(X t) = 0 make for worry; isolated moments of time with F(X t) = ∞ arise for elastic collisions of the hard-wall type but do not invalidate the conclusion.

  6. 6.

    Quantum mechanics of course does not allow for classical nonsense like infinitely fine phase-space structures. Quantum effects smoothen out the classical infinitely fissured landscape just mentioned [7]. In other words, “quantum diffusion” reinforces “Hamiltonian equilibration”.

  7. 7.

    Imagine phase space parametrized by a canonical pair p ∥, q ∥ for momentum and coordinate along the flow, together with f − 1 further pairs making up x ⊥; then replace p ∥ by E and realize \(\frac {\partial E}{\partial p_\|}=\dot {q}_\|\) to conclude dp ∥dq ∥ = dEdt ∥; here dt ∥ equals the time differential dt, but it is well to distinguish the phase-space coordinate confined as 0 ≤ t ∥ < T p from the time t which never ends.

  8. 8.

    There is no contradiction between the propagator being a Dirac delta at all times and equilibration for t →∞; note that the definition of Dirac deltas requires protection by integrals against smooth phase-space densities. See also the reasoning in Sect. 9.11.

  9. 9.

    A focal or conjugate point in configuration space allows for a family of trajectories to fan out, each with a different momentum, which all reunite in another such point; we shall have to deal with conjugate points in the next chapter, see Sect. 10.2.2.

  10. 10.

    To check normalization to unity by integrating over the energy shell would require an extension of the coordinates u, s beyond \({\mathcal {P}}\).

  11. 11.

    In permutation-theory parlance, condition (ii) distinguishes l-encounters for which the permutation 1, 2…, l → i 1, i 2, …, i l of left-port labels to right-port labels is a single cycle, rather than a collection of separate cycles; the cycle structure of permutations will become an issue presently; see Fig. 9.10.

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Authors and Affiliations

  1. Universität Duisburg-Essen, Essen, Germany

    Fritz Haake

  2. School of Mathematical Science, University of Nottingham, Nottingham, UK

    Sven Gnutzmann

  3. Center for Theoretical Physics, Polish Academy of Sciences, Warszawa, Poland

    Marek Kuś

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  1. Fritz Haake
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  2. Sven Gnutzmann
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  3. Marek Kuś
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Haake, F., Gnutzmann, S., Kuś, M. (2018). Classical Hamiltonian Chaos. In: Quantum Signatures of Chaos. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-97580-1_9

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