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The action-angle coordinates for particle motion in magnetic mirror systems

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Abstract

In this paper the problem of the possibility of determining the cyclic Hamiltonian function of a particle moving about a magnetic vessel of the mirror type is discussed. From the Hamiltonian of the typeH=H(P 1,P 2,P 3,Q 3), derived by the author in a previous paper, the Hamiltonian functionH=H(J1,J 2,J 3,w 3) was determined whereP i ,Q i are the generalized impulses and coordinates andJ i ,w k is the action-angle coordinate system. The determination of the form H=H(J1,J 2,J 3) depends on the possibility of solving the quadrature of the Hamilton-Jacobi equation leading to an open form in the general case. Some approximative expressions, suitably replacing the mirror system, are discussed. The solution has been extended to the range of relativistic velocities. The question of the uniqueness of the expression of the Hamiltonian function in cyclic variables is analyzed; it follows from this that the class of canonical transformations leaving the Hamiltonian in a cyclic form does not allow any great simplification of the relatively complicated transformation expressions.

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Krlín, L. The action-angle coordinates for particle motion in magnetic mirror systems. Czech J Phys 17, 124–132 (1967). https://doi.org/10.1007/BF01696119

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Keywords

  • Coordinate System
  • Relativistic Velocity
  • Particle Motion
  • Open Form
  • Approximative Expression