Hamiltonian Structure of the Magnetic Field Lines Ali Khalili Golmankhaneh Seyed Masoud Jazayeri Dumitru Baleanu Iran University of Science Technology, P.O.BOX:6765-63-Tehran-Iran & Physics Department Islamic Azad University, Mahabad Branch, Iran (ali.khalili.gol@gmail.com). Department of Physics, Iran University of Science Technology P.O.BOX:6765-63-Tehran-Iran (jazsm@iust.ac.ir) Cankaya University, 653 Ankara, Turkey Institute of Space Sciences, P.O.BOX, MG-23, R 769, Magurele-Bucharest, Romania (dumitru@cankaya.edu.tr) Abstract: In this paper a geometrical interpretation is given to the divergenceless equations particularly to the magnetic field lines in confined plasmas. By using the geometrical concepts they are formulated in Hamiltonian Nambu formalisms. By writing these equations in the language of the geometrical objects by using the geometrical concepts we could simplify the formalization. As Hamiltonian formalization has advantage of using the mechanical tools, the geometrizing simplifies the calculation the formalization it reveals several deep insights. Keywor: integrable non-integrable systems, forms, Lie derivative, exterior derivative, magnetic surfaces. INTRODUCTION Tokamaks are confined plasma machines that their stabilities are an important problem in which the structure of magnetic field lines plays an fundamental role(; 2; 3; 4; 5; 6; 7; 8; 9; ; ; 2). In addition the plasma confinement in tokamak can be improved by creating a chaotic fiel line region which decreases the plasma-wall interaction. This chaotic region is created by suitably designed resonant magnetic windings which produce the overlapping of two or more chains of magnetic isls, consequently, the magnetic surfaces of destruction. Magnetic fiel lines in general are orbits of hamiltonian systems of one a half degree. Expressing these equation in hamiltonian form has advantage of using powerful metho of hamiltonian mechanics like perturbation theory KAM adiabatic invariance etc(2). The magnetic field line studies indicate the non-integrable divergenceless flows in R 3 may have dense, space filling orbits may show extreme sensibility to initial condition, leading to a loss of predictability, even though the system is deterministic. When a field is represented by means of a magnetic field line Hamiltonian the field lines can be identified with phase space trajectories produced by the Hamiltonian. Mathematically, the issue of magnetic surface reduces to a problem in Hamiltonian mechanics meaning that the whole of mechanics metho is appellable to this problem for example perturbation theory others. The Hamiltonian mechanics itself can be formulated in a geometric way that in this paper we use this formalizations for magnetic field lines. This paper is organized in four sections: In the first section a vector analysis of magnetic field lines is outlined the hamiltonian structure of these equations is derived. This section is divided to subsections. Subsection 2 is devoted to two integrable condition of equations. One of them is analytical way another is numerical. In subsection 3 we discuss the formalization in terms of forms. This method is used to show the Hamiltonian structure of the magnetic field lines. In the next subsection the Lie derivative of forms is applied to show this structure, finally the Poisson bracket method is investigated. In Section 4 we discuss the Nambu structure of the magnetic fiel because this formalization is suitable for describing non-integrable systems. In Section 5 we present our conclusion. 2. HAMILTONIAN FORMULATION 2. Vector analysis The equations of motion of magnetic field lines are given by namical system of equation, ẋ = B, () where B =. When B is a nonlinear function of x namical systems the theory implies that the solutions may exhibit deterministic chaos (e.g. show extreme sensitivity to the initial condition). Integrable divergence less flows in R 3 have only periodic homo clinic or hetero clinic orbits, whose behavior is regular dose not lose predictability over time. We treat two kind of magnetic fiel where the magnetic field lines is restricted to a flux surface another cover ergodically a finite volume. The second field lines are not so much different from the first(near integrable systems). All of the magnetic fiel equations have Hamiltonian structure form. Since B =
, B = A in an arbitrary orthogonal system, ρ, θ, ζ, without assumption of existence of magnetic surfaces, A = A ρ ρ + A θ θ + A ζ ζ, (2) using B = A = (A + G) so A = Ψ θ + Φ ζ + G, (3) where G θ = A θ Ψ, G ζ = A ζ Φ, G ρ = A ρ G = at ρ =. Upon taking the curl, we obtain B = Ψ θ + Φ ζ. (4) (4) contrast with () is called canonical representation. In proper magnetic surface case (intergrade case) the ψ p ψ t is function of ρ. In other wor we have ψ p (ψ t ). But in general case Ψ Φ are dependent on all variables, ρ, θ, ζ, B Φ, B Ψ, Φ(Ψ, θ, ζ). (5) Using B dl =, dl is tangent to the direction of magnetic line, the Hamiltonian form equations is obtained (), Ψ ζ = Φ θ, (6) dζ = Φ Ψ. (7) Considering (4) we obtain dψ = B Ψ = Φ ζ Ψ, (8) = B θ = Φ ζ θ, (9) dζ = B ζ = Ψ θ ζ. () Poisson bracket is define as follows {F, G} = ζ F G = J { f Ψ g θ f θ g }, () Ψ which satisfies antisymmetric condition {f, g} = {g, f} (2) Jacobi identity {f, {g, h}} + {f, {g, h}} + {f, {g, h}} =. (3) The manifold for which the Poisson bracket is defined on it is called Poisson manifold. ζ coordinate is the Casimir invariant of the system since {ζ, f} = ζ ζ f =. (4) The equations of motion in Poisson bracket notation is given by η = {η, H}, (5) where η is replaced with Ψ, θ, ζ, the following equations are obtained dψ = Φ J (6) = Φ J dψ (7) dζ =. (8) Taking J = h(ψ)h (θ), J can be eliminated by taking new variables. For example let us consider the canonical bracket of {f, g} = f g x y g f x y. (9) For the following coordinate transformation we have x = r cos θ (2) y = r sin θ, {f, g} = r (f g r θ f g ). (2) θ r Let J is the Jacobian of (2) choosing J = 2 r2 we obtain {f, g} = f g J θ f g θ J, (22) which is in the stard form. From () the condition of Jacobian is obtained B ζ = J. (23) Consider the magnetic field B = (, B θ (r), B ). (24) Choosing the general method coordinate system r, θ, φ Hamiltonian Φ conjugate momentum Ψ can be found as follows or B θ = Φ φ θ, (25) B θ = Φ r φ θ (26) r R rb θ dr = Bθ dψ = B φ And an equation for Ψ is obtained, Φ dr (27) r Φ dψ. (28) Ψ B φ = Ψ r φ θ, (29) r B φ = Ψ r φ θ, (3) r Ψ rr B φ dr = dr. (3) r It is noticeable that there is no assumption about integrability of system. For this sake the general method is used which is applicable for integrable non-integrable systems. It is interesting to do the above calculations in Cartesian coordinate system, consider A = a x x + a y y + a z z, (32) choosing the direction of A such that a x =,
so ẋ = a y z a z y ẏ = a z x (33) (34) ż = a z y, (35) A = a y y + a z z (36) changing the coordinate system, a z (a y, y, z), we obtain da y = a x y, (37) which are in Hamiltonian structure. = a x a y (38) rational trajectory solutions are periodic enough irrational solutions densely fill the magnetic surfaces..2.2.5.5.5.5 a 2.2 Integrable systems: theoretical numerical method Consider the namical systems which can be written in the following form ẋ = u v. (39) It can be defined a Poisson bracket for these systems as follows {F, G} = u F G. (4) u is the Casimir function because of {u, G} =. (4) The equation of motion is written as follows ẋ = u x v. (42) The systems like (39) are integrable systems. It can be proved that the well-behaved magnetic field lines systems can be written in the following form B = ψ t (θ t ιζ t ), (43) where ι = dψp dψ t. Then, the well-behaved magnetic fiel are integrable systems. Let us Consider the equations ẋ = y ẏ = x, (44) which can be written as ẋ = H z. (45) z component is Casimir variable H is the Hamiltonian of system. Then (44) is integrable. Let us consider the equations, ṙ = θ = ω (46) φ = ω 2, which can be written in the following form ṙ = H(r) (ω θ + ω 2 φ), (47) so this system is integrable. Numerically the integrable systems have regular cross section (Figure,2). All the.2.2.5.5 Fig.. In Cartesian system (x(r,θ, φ),y(r,θ, φ),z(r,θ, φ))a)periodic solution b)enough irrational solutions fills densely the surface. 2 5 5 5 5 2 2 5 5 5 5 2 Fig. 2. (y,z)-plane with cross-section x= c)cross-section of periodic irrational solutions. 2.3 Formalization in terms of forms Consider the following form α = B x + B y dx + B z dx. (48) For the divergenceless forms, dα = ( B x x + B y y + B z )dx =, (49) z we have α = dν. This is the equivalence of vector analysis which says that if B =, B = A..5.5 b c
The Hamiltonian equations can be obtained from the following relation (dp + H q dpdq + H q Equating (48) (5) we obtain H )(dq ) =, (5) p H dq dp =. (5) p B z = J, (52) where J is the Jacobi of transformation from (p, q) to (x, y), x q = p J y y p = q J x Using the identities x p = q J y, (53) y q = p J x. (54) dq = q q dx + x y (55) dp = p p dx + x y (56) (53),(54) we can calculate H, p, q in addition we can define the Poisson bracket as follows {H, f} = H f p q H f q p. (57) 2.4 Hamiltonian vector fiel Along Hamiltonian vector fiel, U, the Lie derivative of ω = dp dq is zero or L U ω =. (58) Changing the coordinate to non-canonical coordinate system ω = dp dq = dx. (59) J Consider the desired equations dx = B x, (6) B z = B y B z. (6) These equations having Hamiltonian structure they must satisfy the equation (58) along the following vector field So, we have From (4) U = d = dx ω(u) = J x + y. (62) dx dx +. (63) J L U (ω) = dω =, (64) dω = x ( B x )dx + J B z y ( B y ) (65) J B z dω = B ln J B ln B z = B ln( J B z ) =. (66) Finally, the condition of such a transformation is obtained B z = J. (67) 2.5 Poisson Brackets, Hamiltonian namics the magnetic field equations Consider the definition of canonical Poisson bracket {f, g} p,q = f g p q f g q p, (68) by transforming to coordinate system (u i ) we have {f, g} u,u 2 = J ( f g u i u j g f u i ). (69) uj From mechanics, a namical system is Hamiltonian if only if the time derivative of PB obeys the following rule(3) d {f, g} = { f, g} + {f, ġ}. (7) The equations,(6),(6), being in Hamiltonian structure they must verify the following relation d {x, y} = = {ẋ, y} + {x, ẏ} = x ( B x ) J B z y ( B y ), (7) J B z using free divergence the relation, B = we lead as a result B ln(j) = B ln(b z ), (72) J = B z. (73) Then, if the equations(6),(6) are in Hamiltonian structure they must satisfy the condition (73). 3. NAMBU STRUCTURE OF THE MAGNETIC FIELD LINES The namical system given by () is obviously non Hamiltonian(the dimension of the phase space is odd). But it is shown that the following volume form is preserved along the magnetic field lines consider, ω = dx (74) ω = dx dx dx + dx + dx dx, (75) where dx = dx dx. Consider vector field B = B x x + B y y + B z z, (76)
from the Lie derivative identity since dω = we have L B (ω) = d(ω(b)) + Bdω (77) L B ω = d(ω(b)), (78) L B ω = (div B)ω. (79) Since div B = or any other divergence-less field, From other side B = dx ω(b) = dx L B ω =. (8) x + + y + z, (8) dx + dx, (82) B x + B y dx + B z dx. (83) From Frobenius theorem equations (83) can be written as B x + B y dx + B z dx = dh dh 2. (84) The condition for the exitances of the function h, h 2 is where Ω = P dx + Q + R (85) Ω dω = P (R y Q z )dx + Q(P z R x ) (86) +R(Q x P y ) =, dω = B. (87) It is important to say that the Nambu method is suitable for describing the non-integrable systems. 4. CONCLUSION In this article we use the geometrical metho for magnetic field lines. The geometric formulation of Nambu mechanics is applied to the magnetic field lines it is found the condition for the existence of the Nambu functions for magnetic field lines. As it is known this formalism is suitable for non-integrable systems. A geometrical formulation has many advantage in namical systems for example enable one to treat some general issues such as integral invariants canonical transformations in a simple way also it gives one deep insights. [3] M. Vlad, F. Spineanu, J. H. Misguich R. Balescu, Magnetic line trapping effective transport in stochastic magnetic fiel Physical Review E 67, 2646 23. [4] R. Balescu Hamiltonian nontwist map for magnetic field lines with locally reversed shear in toroidal geometry. Physical Review E, Vol. 58, No. 3, 998 [5] E. C. da Silva I. L. Caldas Caldas field line diffusion loss in a tokamak with an ergodic magnetic limiter. Physics of Plasmas, Vol. 8, No. 6 [6] V. P. Pastukhov. Nonlinear MHD Convection in Shearless Magnetic System Plasma Physics Reports, Vol. 3, No. 7, 25,57759. [7] S. R. Barocio, E. Chavez-Alarcon, C. Gutierrez- Tapia. Mapping the Intrinsic Stochasticity of Tokamak Divertor Configuration Journal of Physics, Vol. 36, No. 2B, 26 [8] Y. Ishii, M. A Zumi, A. I. Smolyakov. Magnetic isl evolution in rotating plasmas. Journal of Plasma Physics, (26), Vol. 72, part 6, 243248. [9] A. M. Kakurin, I. I. Orlovsky. The Effect of Error Fiel on the Dynamics of a Tearing Mode in a Tokamak, Plasma Physics Reports, Vol. 3, No. 4, 24, 277283. [] A.B. Mikhailovskii. Theory of magnetic isls in tokamaks with accenting neoclassical tearing modes, Plasma Physics, Vol. 43, No. 3-4, 25 77 (23). [] S. R. Huon, J. Breslau. Temperature Contours Ghost Surfaces for Chaotic Magnetic Fiel, Physical Review Letters, PRL, 95 (28) [2] R.L. Viana. Chaotic magnetic filed lines in a tokamak with resonant helical winding, Chaos, Soliton Fractals (2) 765-778 [3] W.D.D Haeseleer, W.N.G.Hitchon, J.D. Callen, J.L.Shohet Flux Coordinate Magnetic Field structure, Springer-Verlag, Berlin, Heildelberg, 99. [4] J. V. Jose, U. J. Saletan. Classical namics:a contemporary approach, Cambridge University Press, 998. [5] V.I. Arnold, A.Avez. Ergodic Problems of Classical Mechanics, Addison-Wesley Publishing Co. Inc. 989. [6] B. F. Schutz. Geometrical metho of mathematical physics, Cambridge University Press, 98. [7] H. Flers. Differential Forms with Application to the Physics, Academic Press, 963. [8] A.J.Lichtenberg, M.A. Lieberman. Regular Chaotic Dynamics, Springer-Verlag, 992. REFERENCES [] I.L. Caldas R.L. Viana. Tokamak magnetic field lines described by simple maps. Europeant Jouenal of Physics, 28 [2] E. C. da Silva, I. E. L. Caldas, Ergodic Magnetic Limiter for the TCABR, Brazilian Journal of Physics, Vol. 32, No., 22