Abstract
The methods developed to study Hamiltonian dynamical systems are also applicable to study magnetic field lines.
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- 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.
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.
Coincidently, the magnetic field of the toroidally confined plasma is a real example of the tori.
- 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.
- 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.
In this section, for the sake of simplicity, we use the notation \(\psi \) for the toroidal flux.
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© 2014 Springer International Publishing Switzerland
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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. https://doi.org/10.1007/978-3-319-01890-4_6
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DOI: https://doi.org/10.1007/978-3-319-01890-4_6
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