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Nonlinear Hamiltonian Systems

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Nonlinear Dynamics and Quantum Chaos

Part of the book series: Graduate Texts in Physics ((GTP))

Abstract

In this chapter we investigate the dynamics of classical nonlinear Hamiltonian systems, which—a priori—are examples of continuous dynamical systems. As in the discrete case (see examples in Chap. 2), we are interested in the classification of their dynamics. After a short review of the basic concepts of Hamiltonian mechanics, we define integrability (and therewith regular motion) in Sect. 3.4. The non-integrability property is then discussed in Sect. 3.5. The addition of small non-integrable parts to the Hamiltonian function (Sects. 3.6.1 and 3.7) leads us to the formal theory of canonical perturbations, which turns out to be a highly valuable technique for the treatment of systems with one degree of freedom and shows profound difficulties when applied to realistic systems with more degrees of freedom. We will interpret these problems as the seeds of chaotic motion in general. A key result for the understanding of the transition from regular to chaotic motion is the KAM theorem (Sect. 3.7.4), which assures the stability in nonlinear systems that are not integrable but behave approximately like them. Within the framework of the surface of section technique, chaotic motion is discussed from a phenomenological point of view in Sect. 3.8. More quantitative measures of local and global chaos are finally presented in Sect. 3.9.

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Notes

  1. 1.

    A separatrix is a boundary in phase space separating two regions of qualitatively different motions, see Fig. 3.1 below for the pendulum.

  2. 2.

    To simplify the notation, vector quantities are not always specified in boldface. From the local context it should be clear that the corresponding symbols represent \(n\)-dimensional vectors.

  3. 3.

    This is due to the fact that the velocity field of \((q,p)\), which is given by \(J \cdot \nabla H(q,p)\) in Eq. (3.2.1) does not depend on time [7].

  4. 4.

    From a mathematical point of view \(Q(q,t)\) has to be a diffeomorphism (differentiable and invertible) for all \(t\).

  5. 5.

    As for the coordinates, the transformation from the old to the new variables has to be a diffeomorphism, see previous footnote.

  6. 6.

    The notion quadrature means the application of basic operations such as addition, multiplication, the computation of inverse functions and the calculation of integrals along a path.

  7. 7.

    The set \(M_C\) is not compact if the motion is not restricted to a finite area in phase space. Examples for such unbounded motion are the free particle or a particle in a linear potential (Wannier-Stark problem).

  8. 8.

    There can be more than one such parameter but for simplicity we assume only one. This result is strictly true only if the angular frequency \(\omega (I,\lambda )\) never equals zero, see [6] and Sect. 3.7.2.

  9. 9.

    This ansatz implicitly takes account of the symplectic structure of the variables \(\eta \) and \(\xi \), i.e. of the area preservation conserved also by the perturbation. The orderly way of doing perturbation theory is formally introduced in the next section.

  10. 10.

    Note that we can assume in Eq. (3.6.26) \(\dot{\theta }\approx \omega \) within this approximation.

  11. 11.

    C.f. footnote in Sect. 3.6.1.

  12. 12.

    Number theory shows that the continued fraction representation is unique [26].

  13. 13.

    The winding number \(\alpha \) depends via the frequencies on both actions \(I_1\) and \(I_2\) but we skip the dependence on \(I_2\) in the following since \(I_2\) is fixed for the Poincaré map, c.f. Eq. (3.8.1).

  14. 14.

    If the map \({\fancyscript{P}}_n\) has a fixed point \(z^*\) there will always be \(n-1\) other fixed points of \({\fancyscript{P}}_n\) given by \({\fancyscript{P}}_i(z^*)\) \((i=1,\ldots ,n-1)\), see Sect. 2.2.1. We say these fixed points belong to the same family.

  15. 15.

    For a short introduction to measures theory consult, e.g., Lieb and Loss [36].

  16. 16.

    For almost every \(\omega \in \Omega \) means for all \(\omega \) except a set of measure zero.

  17. 17.

    To be more precise the points of the trajectory \(z(t)\) are dense on the torus [6].

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  1. Dipartimento di Fisica e Science della Terra, Università di Parma, Parma, Italy

    Sandro Wimberger

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Wimberger, S. (2014). Nonlinear Hamiltonian Systems. In: Nonlinear Dynamics and Quantum Chaos. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06343-0_3

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