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Methods to Study the Hamiltonian Systems

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Part of the Springer Series on Atomic, Optical, and Plasma Physics book series (SSAOPP,volume 78)


The methods developed to study Hamiltonian dynamical systems are also applicable to study magnetic field lines.


  • Hamiltonian System
  • Field Line
  • Magnetic Field Line
  • Magnetic Perturbation
  • Symplectic Mapping

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  • DOI: 10.1007/978-3-319-01890-4_6
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Fig. 6.1
Fig. 6.2
Fig. 6.3
Fig. 6.4
Fig. 6.5
Fig. 6.6
Fig. 6.7
Fig. 6.8
Fig. 6.9
Fig. 6.10
Fig. 6.11


  1. 1.

    The definition of integrable of Hamiltonian systems is given by the Liouville’s theorem. Its modern formulation can be found in Arnold (1989).

  2. 2.

    As was noted in Sec. 5.4 in some Hamiltonian systems with degrees of freedom \(N\ge 2\) one cannot introduce global action–angle variables in whole phase space. Such a phenomenon is known as Hamiltonian monodromy (Duistermaat (1980)).

  3. 3.

    Coincidently, the magnetic field of the toroidally confined plasma is a real example of the tori.

  4. 4.

    The word “symplectic” comes from Greek meaning “twining or plaiting together”, since the canonical coordinates \(q_i\) and momenta \(p_i\) are intertwined by symplectic 2-form given as a sum of wedge products of \(dp_i\) with \(dq_i\), \(\omega ^2= \sum _{i=1}^N dq_i \wedge dp_i=\sum _{i=1}^N dI_i\wedge d\vartheta _i\). This structure is invariant under the Hamiltonian flow given by Eq. (6.8) (see, e.g. Arnold 1989). The wedge product \(dq_{i} \wedge dp_{i}\) is equal to an area of a parallelogram with sides \(dq_{i}\) and \(dp_{i}\).

  5. 5.

    Symplectic mapping method for integrating the Hamiltonian system (6.7) and (6.8) based on the method of canonical transformation of variables and the classical perturbation theory has been developed by Abdullaev et al. (1999); Abdullaev (1999, 2002, 2004a) (see also a book by Abdullaev (2006)).

  6. 6.

    Interestingly that this integral has been introduced to study the magnetic field lines produced by the helical current flowing the surface of a torus (Melnikov (1963a)).

  7. 7.

    In this section, for the sake of simplicity, we use the notation \(\psi \) for the toroidal flux.

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Correspondence to Sadrilla Abdullaev .

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Abdullaev, S. (2014). Methods to Study the Hamiltonian Systems. In: Magnetic Stochasticity in Magnetically Confined Fusion Plasmas. Springer Series on Atomic, Optical, and Plasma Physics, vol 78. Springer, Cham.

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