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Hamiltonian Representation of Magnetic Field

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

Abstract

In this chapter we present the Hamiltonian formulation of the equations for magnetic field lines. We specifically consider the magnetic field corresponding the toroidal plasma configuration.

Keywords

  • Toroidal Plasma Configuration
  • Field Lines
  • Flux Coordinates
  • Field Line Equations
  • Clebsch Form

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Notes

  1. 1.

    A diamagnetic current is created due to a circular motion of charged particles in an induced magnetic field. It produces a magnetic field which opposes the external magnetic field and thus the total magnetic field is reduced. Therefore a plasma possesses diamagnetic properties. Typically, the magnetic field due to a diamagnetic current is much smaller than the strong toroidal field \(B_0\).

  2. 2.

    The magnetic flux \(\psi \) cannot be arbitrary small. According to the quantization rule, \(\oint p_zdz=h n\), (\(n=1,2,\dots \)), \(p_z= eA/c\) one has \(\psi = \varPhi _0 n\), where \(\varPhi _0=hc/e\) is a quant of magnetic flux [\(h\) is the Planck’s constant]. In fusion plasmas the magnetic flux is so large, that its discreteness does not play any role.

  3. 3.

    Particularly, as will be discussed Chap. 11 the barriers to a particle transport caused by a small scale turbulent field may be formed near the low–order rational values of \(q\).

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Correspondence to Sadrilla Abdullaev .

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Abdullaev, S. (2014). Hamiltonian Representation of Magnetic Field. 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_1

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