- Email: [email protected]
Structure of magnetic field lines
Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Structure of magnetic field lines Ali Khalili Golmankhaneh a,⇑, Alireza Khalili Golmankhaneh b, Seyed Masud Jazayeri c, Dumitru Baleanu d,e a
Department of Physics, Islamic Azad University, Mahabad Branch, Mahabad, Iran Department of Physics, Islamic Azad University, Urmia Branch, P.O. Box 969, Uromiyeh, Iran c Department of Physics, University of Iran Science and Technology, P.O. Box 16765-163, Tehran, Iran d Department of Mathematics and Computer Science, Cankaya University, 06530 Ankara, Turkey e Institute of Space Sciences, P.O. Box MG-23, R 76900 Magurele-Bucharest, Romaniab b
a r t i c l e
i n f o
Article history: Available online 6 April 2011 Keywords: Integrable systems Non-integrable systems Forms Lie derivative Exterior derivative Magnetic surfaces Nambu mechanics
a b s t r a c t In this paper the Hamiltonian structure of magnetic lines is studied in many ways. First it is used vector analysis for defining the Poisson bracket and Casimir variable for this system. Second it is derived Pfaffian equations for magnetic field lines. Third, Lie derivative and derivative of Poisson bracket is used to show structure of this system. Finally, it is shown Nambu structure of the magnetic field lines. Ó 2011 Elsevier B.V. All rights reserved.
1. 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 [1–18]. In addition the plasma confinement in Tokamak can be improved by creating a chaotic fields 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 islands and, consequently, the magnetic surfaces of destruction. Magnetic fields lines in general are orbits of Hamiltonian systems of one and a half degree. Expressing these equation in Hamiltonian form has advantage of using powerful methods of Hamiltonian mechanics like perturbation theory and KAM and adiabatic invariance and etc. [12]. The magnetic field line studies indicate the non-integrable divergenceless flows in R3 may have dense, and space filling orbits and 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 methods is appellable to this problem for example perturbation theory and 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 as follows: In Section 2 we have studied the structure of magnetic magnetic field lines using vector analysis, Pfaffian equations, Lie derivative and derivative of Poisson bracket. Section 3 is devoted to explain Nambu structure of the magnetic field lines. In last it is given a short conclusion. ⇑ Corresponding author. E-mail addresses: [email protected] (A.K. Golmankhaneh), [email protected] (A.K. Golmankhaneh), [email protected] (S.M. Jazayeri), [email protected] (D. Baleanu). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.03.042
714
A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
2. Hamiltonian structure of magnetic field lines 2.1. Vector analysis The equations of motion of magnetic field lines is given by dynamical system of equation,
r_ ¼ B;
ð1Þ
where r B = 0. When B is nonlinear function of r dynamical systems theory implies that the solutions may exhibit deterministic chaos (e.g. show extreme sensitivity to initial condition). Integrable divergence less flows in R3 have only periodic and homo clinic or hetero clinic orbits, whose behavior is regular and dose not lose predictability over time. We treat two kind of magnetic fields where magnetic field lines restricted to a flux surface and another cover ergodically a finite volume. The second field lines are not so much differ from the first (near integrable systems). All of the magnetic fields equations have Hamiltonian structure form. Since r B = 0, B = r A in an arbitrary orthogonal system, q, h, f, without assumption of existence of magnetic surfaces,
A ¼ Aq rq þ Ah rh þ Af rf;
ð2Þ
using B = r A = r (A + rG) so
A ¼ Wrh þ Urf þ rG; where
@G @q
ð3Þ
¼ Aq , W ¼ Ah @G , U ¼ Af þ @G , and G = 0 at q = 0. Taking the curl of Eq. (3), we obtain @h @f
B ¼ rW rh þ rU rf:
ð4Þ
Eq. (4) contrast with Eq. (1) is called canonical representation. In proper magnetic surface case the W and U are functions of q, also we could write W(U). But in general case W and U are dependent on all variables, q, h, f, B rU – 0, B rW – 0, and
UðW; h; fÞ:
ð5Þ
Using B dl = 0, where dl is tangent to the direction of magnetic line, the equations in the Hamilton form are obtained as follows [1],
@W @U ¼ ; @f @h dh @U : ¼ df @W
ð6Þ ð7Þ
Considering Eq. (4) and using dv = rv dr and ddsv ¼ rv dr , we have ds
dW ¼ B rW ¼ rU rf rW; ds dh ¼ B rh ¼ rU rf rh; ds df ¼ B rf ¼ rW rh rf; ds where
dr ds
ð8Þ ð9Þ ð10Þ
¼ B. Now the Poisson bracket is define as follows:
ff ; gg ¼ rf rf rg ¼
1 @f @g @f @g ; J @ W @h @h @ W
ð11Þ
which satisfies antisymmetric condition
ff ; gg ¼ fg; f g
ð12Þ
and Jacobi identity
ff ; fg; hgg þ ff ; fg; hgg þ ff ; fg; hgg ¼ 0:
ð13Þ
The manifold which the Poisson bracket is defined on it is called Poisson manifold. f coordinate is the Casimir invariant of the system since
ff; f g ¼ rf rf rf ¼ 0:
ð14Þ
The equations of motion in Poisson bracket notation is given by
g_ ¼ fg; Hg;
ð15Þ
A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
715
where g is replaced with W, h, then the following equations are obtained
dW 1 @ U ¼ ; J dh ds dh 1 @ U ¼ : ds J dW
ð16Þ ð17Þ
Taking J = h(W)h0 (h), J can be eliminated by taking new variables, for example: consider the canonical bracket,
ff ; gg ¼
@f @g @g @f : @x @y @x @y
ð18Þ
For the following coordinate transformation
x ¼ q cos h;
ð19Þ
y ¼ q sin h the bracket changes to
ff ; gg ¼
1
@f @g @f @g q @ q @h @h @ q
ð20Þ
qis the Jacobian of Eq. (19), by choosing J ¼ 12 q2 we have ff ; gg ¼
@f @g @f @g @J @h @h @J
ð21Þ
which is in the standard form. From Eq. (10) the condition of Jacobian is obtained as
Bf ¼
1 : J
ð22Þ
2.1.1. Example Consider the magnetic field of stationary state of long ratio Tokamak (where q is small radius and R0 is large one)
B ¼ ð0; Bh ðqÞ; B0 Þ:
ð23Þ
By choosing coordinate system q, h, f the Hamiltonian, U, and conjugate momentum, W, can be found as follows:
B rh ¼ rU rf rh;
ð24Þ
then
B rh ¼
@U rq r/ rh @q
rq r/ rh ¼ R01q Z Z @U dq R0 qBh dq ¼ @q
ð25Þ
ð26Þ
or
Z
Bh dW ¼ Bf
Z
@U dW; @W
ð27Þ
whererW rf rh = Bf. Now the equation for W is obtained as follows:
B rf ¼
@W rq r/ rh; @q
ð28Þ
and
Bf ¼
@W rq r/ rh; @q
ð29Þ
and using rq r/ rh ¼ R01q
Z
R0 qB/ dq ¼
Z
@W dq: @q
ð30Þ
It is interesting to do above calculations in cartesian coordinate system, then consider
A ¼ ax rx þ ay ry þ az rz;
ð31Þ
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A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
choosing the direction of A such that ax = 0, then
@ay @az ; @z @y @az ; y_ ¼ @x @ay ; z_ ¼ @x
x_ ¼
ð32Þ ð33Þ ð34Þ
so
r A ¼ ray ry þ raz rz
ð35Þ
it is mentioned that this kind of field could be written in Hamiltonian structure where U = az andW = ay then we have
day @ax ¼ ; dz @y dy @ax ¼ ; dz @ay
ð36Þ ð37Þ
which are in Hamiltonian structure. 2.2. An integrable system Consider the dynamical systems which can be written in the following form
r_ ¼ ru rv :
ð38Þ
It can be defined Poisson bracket for these systems as follows:
fF; Gg ¼ ru rF rG:
ð39Þ
u will be the Casimir function because of
fu; Gg ¼ 0:
ð40Þ
And the equation of motion is written as follows:
x_i ¼ ru rxi rv :
ð41Þ
The systems, like Eq. (38), are integrable systems. It can be proved that well-behave magnetic field lines systems can be written in the following form
B ¼ rU rðh ifÞ; where
i¼
dU . dW
ð42Þ
Then the well-behaved magnetic fields are integrable systems.
2.2.1. Example Consider the equations
x_ ¼ y;
y_ ¼ x;
ð43Þ
then it can be written as
q_ ¼ rH rz;
ð44Þ
where z component is the Casimir variable and H is the Hamiltonian of system. As a result Eq. (43) is integrable. 2.2.2. Example Consider the equations,
r_ ¼ 0; h_ ¼ x1 ;
ð45Þ
/_ ¼ x2 ; which can be written in the following form
r_ ¼ rHðrÞ rðx1 h þ x2 /Þ;
ð46Þ
so this system is integrable. Numerically the integrable systems have regular cross section (see Figs. 1, 2). All the rational trajectory solutions are periodic and enough irrational solutions densely fill the magnetic surfaces.
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A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
a
b 0.2 0 −0.2
0.2 0 −0.2 1
1
0.5
0 −0.5 −1
−1
−0.5
0
0.5
1
0.5
0
−0.5
−1
−1
−0.5
0
0.5
1
Fig. 1. In cartesian system (x(r, h, /), y(r, h, /), z(r, h, /)) (a) periodic solution and (b) enough irrational solutions fills densely the surface.
2.3. Forms formalization Consider the following form
a ¼ Bx dy dz þ By dx dz þ Bz dx dy:
ð47Þ
For the divergence-less forms,
da ¼
@Bx @By @Bz þ þ dx dy dz ¼ 0; @x @y @z
ð48Þ
so a = dm. This is the equivalence of vector analysis which says that if r B = 0, then B = r A. The Hamiltonian equations can be obtained from the following relation
@H @H dt dq dt ¼ 0; dp þ @q @p
ð49Þ
then
dpdq þ
@H @H dtdq dpdt ¼ 0: @q @p
ð50Þ
Equating Eqs. (47) and (50), then we obtain
Bz ¼ J;
ð51Þ
where J is the Jacobi of transformation from (p,q) to (x,y), and
@x 1 ¼ @q J @y 1 ¼ @p J
@p ; @y @q ; @x
@x 1 ¼ @p J @y 1 ¼ @q J
@q ; @y @p : @x
ð52Þ ð53Þ
Using the identities
@q @q dx þ dy; @x @y @p @p dp ¼ dx þ dy @x @y
dq ¼
ð54Þ ð55Þ
and Eqs. (52) and (53) we can calculate H, p, q and in addition we can define the Poisson bracket as follows:
fH; f g ¼
@H @f @H @f : @p @q @q @p
ð56Þ
2.4. Hamiltonian vector fields Along the Hamiltonian vector field, U, the Lie derivative of x = dp ^ dq is zero or
LU x ¼ 0:
ð57Þ
718
A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720 20
15
10
5
0
−5
−10
−15
−20
-20
-15
-10
-5
0
5
10
15
20
c
Fig. 2. (y, z)-plane with cross-section x = 0 (c) cross-section of periodic and irrational solutions.
Changing the coordinate to non-canonical coordinate system, then we have
1 J
x ¼ dp ^ dq ¼ dx ^ dy:
ð58Þ
Consider the desired equations as
dx Bx ¼ ; dz Bz dy By ¼ : dz Bz
ð59Þ ð60Þ
These equations having a Hamiltonian structure must satisfy Eq. (57) along the following vector field
U¼
d dx @ dy @ ¼ þ : dz dz @x dz @y
ð61Þ
So we have
1 dx 1 dy dx þ dy: J dz J dz
xðUÞ ¼
ð62Þ
From [4]
LU ðxÞ ¼ dx ¼ 0;
ð63Þ
then
dx ¼
@ 1 Bx @ 1 By þ dx ^ dy @x J Bz @y J Bz
ð64Þ
and
B r ln J B r ln Bz ¼ B r ln
J ¼ 0: Bz
ð65Þ
Finally, the condition of such a transformation is obtained as
Bz ¼ J:
ð66Þ
A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
719
2.5. Poisson bracket, Hamiltonian dynamics and magnetic field equations Consider the definition of canonical Poisson bracket as given below:
ff ; ggp;q ¼
@f @g @f @g : @p @q @q @p
ð67Þ
By transforming to the coordinate system (ui) we have
ff ; ggu1 ;u2 ¼
1 @f @g @g @f : i i j j J @u @u @u @u
ð68Þ
As it is known from mechanics, a dynamical system is Hamiltonian if and only if the time derivative of PB obeys the following rule [3]
d _ ff ; gg ¼ ff_ ; gg þ ff ; gg: dt
ð69Þ
Eqs. (59) and (60) being in a Hamiltonian structure, they must verify the following relation
d @ 1 Bx @ 1 By _ yg þ fx; yg _ ¼ ; fx; yg ¼ 0 ¼ fx; dz @x J Bz @y J Bz
ð70Þ
and by using free divergence the relation, r B = 0 we lead
B r lnðJÞ ¼ B r lnðBz Þ:
ð71Þ
As a result we obtain
J ¼ Bz :
ð72Þ
Then, if Eqs. (59) and (60) are in a Hamiltonian structure they must satisfy the condition equation (72). 3. Nambu structure of the magnetic field lines The dynamical system given by Eq. (1) is obviously a 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. Let us consider
x ¼ dx ^ dy ^ dz
ð73Þ
x ¼ dx dy dz dx dz dy dy dx dz þ dy dz dx þ dz dy dx dz dx dy;
ð74Þ
and,
where dx ^ dy = dx dy dy dx. In addition we Consider the vector field as
B ¼ Bx
@ @ @ þ By þ Bz : @x @y @z
ð75Þ
From the Lie derivative identity
LB ðxÞ ¼ dðxðBÞÞ þ Bdx
ð76Þ
and since dx = 0 we obtain that
LB x ¼ dðxðBÞÞ;
ð77Þ
therefore
LB x ¼ ðdivBÞx:
ð78Þ
Since div B = 0 or any other divergenceless field,
LB x ¼ 0:
ð79Þ
On the other side we have
B¼
dx @ dy @ dz @ þ þ ; dt @x dt @y dt @z
ð80Þ
then
xðBÞ ¼
dx dy dz dy dz þ dx dz þ dx dz; dt dt dt
ð81Þ
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A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
and
Bx dy dz þ By dx dz þ Bz dx dy:
ð82Þ
By using the Frobenius theorem Eq. (82) can be written as
Bx dy dz þ By dx dz þ Bz dx dy ¼ dh1 ^ dh2 :
ð83Þ
The condition for the existence of the function h1, h2 is
X ¼ Pdx þ Qdy þ Rdz
ð84Þ
X ^ dX ¼ PðRy Q z Þdx þ Q ðPz Rx Þdy þ RðQ x Py Þdz ¼ 0;
ð85Þ
dX ¼ B:
ð86Þ
and
where
We notice that the Nambu method is suitable for describing the non-integrable systems. 4. Conclusion Formalization of the magnetic field in Hamiltonian form has advantage of using the mechanical tools as KAM and perturbation theory. For example due to the Hamiltonian character of the magnetic lines the problem of the distributive instabilities can be considered from the point of the canonical perturbation theory. Each equation maybe has many geometric conception. Hamiltonian equations also could be obtain from the geometric ways. In this article we use this method for the magnetic field lines. Also, the geometric formulation of Nambu mechanics is applied to the magnetic field lines and it is found the condition for the existence of Nambu functions for the magnetic field lines.This formalism is suitable for non-integrable systems. In addition of this, a geometrical formulation has many advantages within the dynamical systems, for example, enable us to treat some general issues such as integral invariants and canonical transformations in simple way and also gives one deep insights. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Caldas IL, Viana RL. Tokamak magnetic field lines described by simple maps. Eur Phys J Special Top 2008;165:195–210. da Silva EC, Caldas IL. Ergodic magnetic limiter for the TCABR. Braz J Phys 2002;32:39–45. Vlad M, Spineanu F, Misguich JH, Balescu R. Magnetic line trapping and effective transport in stochastic magnetic fields. Phys Rev E 2003;67:026406. Balescu R. Hamiltonian nontwist map for magnetic field lines with locally reversed shear in toroidal geometry. Phys Rev E 1998;58:3781–92. da Silva EC, Caldas Ibere L. Field line diffusion and loss in a tokamak with an ergodic magnetic limiter. Phys Plasmas 2001;8:2855–65. Pastukhov VP. Nonlinear MHD convection in shearless magnetic system. Plasma Phys Rep 2005;31:577–90. Barocio SR, Chavez-Alarcon E, Gutierrez-Tapia C. Mapping the intrinsic stochasticity of Tokamak divertor configuration. Braz J Phys 2006;36:550–6. Ishii Y, Zumi MA, Smolyakov AI. Magnetic island evolution in rotating plasmas. J Plasma Phys 2006;72:1243–8. Kakurin AM, Orlovsky II. The effect of error fields on the dynamics of a tearing mode in a Tokamak. Plasma Phys Rep 2004;30:277–83. Mikhailovskii AB. Theory of magnetic islands in Tokamaks with accenting neoclassical tearing modes. Plasma Phys 2003;43:125177. Hudson SR, Breslau J. Temperature contours and ghost surfaces for chaotic magnetic fields. Phys Rev Lett 2008;100:095001. Viana RL. Chaotic magnetic filed lines in a Tokamak with resonant helical winding. Chaos Soliton Fract 2000;11:765–78. D’haeseleer WD, Hitchon WNG, Callen JD, Shohet JL. Flux coordinate and magnetic field structure. Berlin, Heidelberg: Springer-Verlag; 1991. Jose JV, Saletan UJ. Classical dynamics: a contemporary approach. Cambridge University Press; 1998. Arnold VI, Avez A. Ergodic problems of classical mechanics. Math. physics monograph series, Benjamin, NY; 1968. Schutz BF. Geometrical methods of mathematical physics. Cambridge University Press; 1980. Flanders H. Differential forms with application to the physical sciences. NY: Academic Press; 1963. Lichtenberg AJ, Lieberman MA. Regular and chaotic dynamics. NY: Springer-Verlag; 1992.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Structure of magnetic field lines Ali Khalili Golmankhaneh a,⇑, Alireza Khalili Golmankhaneh b, Seyed Masud Jazayeri c, Dumitru Baleanu d,e a
Department of Physics, Islamic Azad University, Mahabad Branch, Mahabad, Iran Department of Physics, Islamic Azad University, Urmia Branch, P.O. Box 969, Uromiyeh, Iran c Department of Physics, University of Iran Science and Technology, P.O. Box 16765-163, Tehran, Iran d Department of Mathematics and Computer Science, Cankaya University, 06530 Ankara, Turkey e Institute of Space Sciences, P.O. Box MG-23, R 76900 Magurele-Bucharest, Romaniab b
a r t i c l e
i n f o
Article history: Available online 6 April 2011 Keywords: Integrable systems Non-integrable systems Forms Lie derivative Exterior derivative Magnetic surfaces Nambu mechanics
a b s t r a c t In this paper the Hamiltonian structure of magnetic lines is studied in many ways. First it is used vector analysis for defining the Poisson bracket and Casimir variable for this system. Second it is derived Pfaffian equations for magnetic field lines. Third, Lie derivative and derivative of Poisson bracket is used to show structure of this system. Finally, it is shown Nambu structure of the magnetic field lines. Ó 2011 Elsevier B.V. All rights reserved.
1. 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 [1–18]. In addition the plasma confinement in Tokamak can be improved by creating a chaotic fields 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 islands and, consequently, the magnetic surfaces of destruction. Magnetic fields lines in general are orbits of Hamiltonian systems of one and a half degree. Expressing these equation in Hamiltonian form has advantage of using powerful methods of Hamiltonian mechanics like perturbation theory and KAM and adiabatic invariance and etc. [12]. The magnetic field line studies indicate the non-integrable divergenceless flows in R3 may have dense, and space filling orbits and 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 methods is appellable to this problem for example perturbation theory and 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 as follows: In Section 2 we have studied the structure of magnetic magnetic field lines using vector analysis, Pfaffian equations, Lie derivative and derivative of Poisson bracket. Section 3 is devoted to explain Nambu structure of the magnetic field lines. In last it is given a short conclusion. ⇑ Corresponding author. E-mail addresses: [email protected] (A.K. Golmankhaneh), [email protected] (A.K. Golmankhaneh), [email protected] (S.M. Jazayeri), [email protected] (D. Baleanu). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.03.042
714
A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
2. Hamiltonian structure of magnetic field lines 2.1. Vector analysis The equations of motion of magnetic field lines is given by dynamical system of equation,
r_ ¼ B;
ð1Þ
where r B = 0. When B is nonlinear function of r dynamical systems theory implies that the solutions may exhibit deterministic chaos (e.g. show extreme sensitivity to initial condition). Integrable divergence less flows in R3 have only periodic and homo clinic or hetero clinic orbits, whose behavior is regular and dose not lose predictability over time. We treat two kind of magnetic fields where magnetic field lines restricted to a flux surface and another cover ergodically a finite volume. The second field lines are not so much differ from the first (near integrable systems). All of the magnetic fields equations have Hamiltonian structure form. Since r B = 0, B = r A in an arbitrary orthogonal system, q, h, f, without assumption of existence of magnetic surfaces,
A ¼ Aq rq þ Ah rh þ Af rf;
ð2Þ
using B = r A = r (A + rG) so
A ¼ Wrh þ Urf þ rG; where
@G @q
ð3Þ
¼ Aq , W ¼ Ah @G , U ¼ Af þ @G , and G = 0 at q = 0. Taking the curl of Eq. (3), we obtain @h @f
B ¼ rW rh þ rU rf:
ð4Þ
Eq. (4) contrast with Eq. (1) is called canonical representation. In proper magnetic surface case the W and U are functions of q, also we could write W(U). But in general case W and U are dependent on all variables, q, h, f, B rU – 0, B rW – 0, and
UðW; h; fÞ:
ð5Þ
Using B dl = 0, where dl is tangent to the direction of magnetic line, the equations in the Hamilton form are obtained as follows [1],
@W @U ¼ ; @f @h dh @U : ¼ df @W
ð6Þ ð7Þ
Considering Eq. (4) and using dv = rv dr and ddsv ¼ rv dr , we have ds
dW ¼ B rW ¼ rU rf rW; ds dh ¼ B rh ¼ rU rf rh; ds df ¼ B rf ¼ rW rh rf; ds where
dr ds
ð8Þ ð9Þ ð10Þ
¼ B. Now the Poisson bracket is define as follows:
ff ; gg ¼ rf rf rg ¼
1 @f @g @f @g ; J @ W @h @h @ W
ð11Þ
which satisfies antisymmetric condition
ff ; gg ¼ fg; f g
ð12Þ
and Jacobi identity
ff ; fg; hgg þ ff ; fg; hgg þ ff ; fg; hgg ¼ 0:
ð13Þ
The manifold which the Poisson bracket is defined on it is called Poisson manifold. f coordinate is the Casimir invariant of the system since
ff; f g ¼ rf rf rf ¼ 0:
ð14Þ
The equations of motion in Poisson bracket notation is given by
g_ ¼ fg; Hg;
ð15Þ
A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
715
where g is replaced with W, h, then the following equations are obtained
dW 1 @ U ¼ ; J dh ds dh 1 @ U ¼ : ds J dW
ð16Þ ð17Þ
Taking J = h(W)h0 (h), J can be eliminated by taking new variables, for example: consider the canonical bracket,
ff ; gg ¼
@f @g @g @f : @x @y @x @y
ð18Þ
For the following coordinate transformation
x ¼ q cos h;
ð19Þ
y ¼ q sin h the bracket changes to
ff ; gg ¼
1
@f @g @f @g q @ q @h @h @ q
ð20Þ
qis the Jacobian of Eq. (19), by choosing J ¼ 12 q2 we have ff ; gg ¼
@f @g @f @g @J @h @h @J
ð21Þ
which is in the standard form. From Eq. (10) the condition of Jacobian is obtained as
Bf ¼
1 : J
ð22Þ
2.1.1. Example Consider the magnetic field of stationary state of long ratio Tokamak (where q is small radius and R0 is large one)
B ¼ ð0; Bh ðqÞ; B0 Þ:
ð23Þ
By choosing coordinate system q, h, f the Hamiltonian, U, and conjugate momentum, W, can be found as follows:
B rh ¼ rU rf rh;
ð24Þ
then
B rh ¼
@U rq r/ rh @q
rq r/ rh ¼ R01q Z Z @U dq R0 qBh dq ¼ @q
ð25Þ
ð26Þ
or
Z
Bh dW ¼ Bf
Z
@U dW; @W
ð27Þ
whererW rf rh = Bf. Now the equation for W is obtained as follows:
B rf ¼
@W rq r/ rh; @q
ð28Þ
and
Bf ¼
@W rq r/ rh; @q
ð29Þ
and using rq r/ rh ¼ R01q
Z
R0 qB/ dq ¼
Z
@W dq: @q
ð30Þ
It is interesting to do above calculations in cartesian coordinate system, then consider
A ¼ ax rx þ ay ry þ az rz;
ð31Þ
716
A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
choosing the direction of A such that ax = 0, then
@ay @az ; @z @y @az ; y_ ¼ @x @ay ; z_ ¼ @x
x_ ¼
ð32Þ ð33Þ ð34Þ
so
r A ¼ ray ry þ raz rz
ð35Þ
it is mentioned that this kind of field could be written in Hamiltonian structure where U = az andW = ay then we have
day @ax ¼ ; dz @y dy @ax ¼ ; dz @ay
ð36Þ ð37Þ
which are in Hamiltonian structure. 2.2. An integrable system Consider the dynamical systems which can be written in the following form
r_ ¼ ru rv :
ð38Þ
It can be defined Poisson bracket for these systems as follows:
fF; Gg ¼ ru rF rG:
ð39Þ
u will be the Casimir function because of
fu; Gg ¼ 0:
ð40Þ
And the equation of motion is written as follows:
x_i ¼ ru rxi rv :
ð41Þ
The systems, like Eq. (38), are integrable systems. It can be proved that well-behave magnetic field lines systems can be written in the following form
B ¼ rU rðh ifÞ; where
i¼
dU . dW
ð42Þ
Then the well-behaved magnetic fields are integrable systems.
2.2.1. Example Consider the equations
x_ ¼ y;
y_ ¼ x;
ð43Þ
then it can be written as
q_ ¼ rH rz;
ð44Þ
where z component is the Casimir variable and H is the Hamiltonian of system. As a result Eq. (43) is integrable. 2.2.2. Example Consider the equations,
r_ ¼ 0; h_ ¼ x1 ;
ð45Þ
/_ ¼ x2 ; which can be written in the following form
r_ ¼ rHðrÞ rðx1 h þ x2 /Þ;
ð46Þ
so this system is integrable. Numerically the integrable systems have regular cross section (see Figs. 1, 2). All the rational trajectory solutions are periodic and enough irrational solutions densely fill the magnetic surfaces.
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a
b 0.2 0 −0.2
0.2 0 −0.2 1
1
0.5
0 −0.5 −1
−1
−0.5
0
0.5
1
0.5
0
−0.5
−1
−1
−0.5
0
0.5
1
Fig. 1. In cartesian system (x(r, h, /), y(r, h, /), z(r, h, /)) (a) periodic solution and (b) enough irrational solutions fills densely the surface.
2.3. Forms formalization Consider the following form
a ¼ Bx dy dz þ By dx dz þ Bz dx dy:
ð47Þ
For the divergence-less forms,
da ¼
@Bx @By @Bz þ þ dx dy dz ¼ 0; @x @y @z
ð48Þ
so a = dm. This is the equivalence of vector analysis which says that if r B = 0, then B = r A. The Hamiltonian equations can be obtained from the following relation
@H @H dt dq dt ¼ 0; dp þ @q @p
ð49Þ
then
dpdq þ
@H @H dtdq dpdt ¼ 0: @q @p
ð50Þ
Equating Eqs. (47) and (50), then we obtain
Bz ¼ J;
ð51Þ
where J is the Jacobi of transformation from (p,q) to (x,y), and
@x 1 ¼ @q J @y 1 ¼ @p J
@p ; @y @q ; @x
@x 1 ¼ @p J @y 1 ¼ @q J
@q ; @y @p : @x
ð52Þ ð53Þ
Using the identities
@q @q dx þ dy; @x @y @p @p dp ¼ dx þ dy @x @y
dq ¼
ð54Þ ð55Þ
and Eqs. (52) and (53) we can calculate H, p, q and in addition we can define the Poisson bracket as follows:
fH; f g ¼
@H @f @H @f : @p @q @q @p
ð56Þ
2.4. Hamiltonian vector fields Along the Hamiltonian vector field, U, the Lie derivative of x = dp ^ dq is zero or
LU x ¼ 0:
ð57Þ
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15
10
5
0
−5
−10
−15
−20
-20
-15
-10
-5
0
5
10
15
20
c
Fig. 2. (y, z)-plane with cross-section x = 0 (c) cross-section of periodic and irrational solutions.
Changing the coordinate to non-canonical coordinate system, then we have
1 J
x ¼ dp ^ dq ¼ dx ^ dy:
ð58Þ
Consider the desired equations as
dx Bx ¼ ; dz Bz dy By ¼ : dz Bz
ð59Þ ð60Þ
These equations having a Hamiltonian structure must satisfy Eq. (57) along the following vector field
U¼
d dx @ dy @ ¼ þ : dz dz @x dz @y
ð61Þ
So we have
1 dx 1 dy dx þ dy: J dz J dz
xðUÞ ¼
ð62Þ
From [4]
LU ðxÞ ¼ dx ¼ 0;
ð63Þ
then
dx ¼
@ 1 Bx @ 1 By þ dx ^ dy @x J Bz @y J Bz
ð64Þ
and
B r ln J B r ln Bz ¼ B r ln
J ¼ 0: Bz
ð65Þ
Finally, the condition of such a transformation is obtained as
Bz ¼ J:
ð66Þ
A.K. Golmankhaneh et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 713–720
719
2.5. Poisson bracket, Hamiltonian dynamics and magnetic field equations Consider the definition of canonical Poisson bracket as given below:
ff ; ggp;q ¼
@f @g @f @g : @p @q @q @p
ð67Þ
By transforming to the coordinate system (ui) we have
ff ; ggu1 ;u2 ¼
1 @f @g @g @f : i i j j J @u @u @u @u
ð68Þ
As it is known from mechanics, a dynamical system is Hamiltonian if and only if the time derivative of PB obeys the following rule [3]
d _ ff ; gg ¼ ff_ ; gg þ ff ; gg: dt
ð69Þ
Eqs. (59) and (60) being in a Hamiltonian structure, they must verify the following relation
d @ 1 Bx @ 1 By _ yg þ fx; yg _ ¼ ; fx; yg ¼ 0 ¼ fx; dz @x J Bz @y J Bz
ð70Þ
and by using free divergence the relation, r B = 0 we lead
B r lnðJÞ ¼ B r lnðBz Þ:
ð71Þ
As a result we obtain
J ¼ Bz :
ð72Þ
Then, if Eqs. (59) and (60) are in a Hamiltonian structure they must satisfy the condition equation (72). 3. Nambu structure of the magnetic field lines The dynamical system given by Eq. (1) is obviously a 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. Let us consider
x ¼ dx ^ dy ^ dz
ð73Þ
x ¼ dx dy dz dx dz dy dy dx dz þ dy dz dx þ dz dy dx dz dx dy;
ð74Þ
and,
where dx ^ dy = dx dy dy dx. In addition we Consider the vector field as
B ¼ Bx
@ @ @ þ By þ Bz : @x @y @z
ð75Þ
From the Lie derivative identity
LB ðxÞ ¼ dðxðBÞÞ þ Bdx
ð76Þ
and since dx = 0 we obtain that
LB x ¼ dðxðBÞÞ;
ð77Þ
therefore
LB x ¼ ðdivBÞx:
ð78Þ
Since div B = 0 or any other divergenceless field,
LB x ¼ 0:
ð79Þ
On the other side we have
B¼
dx @ dy @ dz @ þ þ ; dt @x dt @y dt @z
ð80Þ
then
xðBÞ ¼
dx dy dz dy dz þ dx dz þ dx dz; dt dt dt
ð81Þ
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and
Bx dy dz þ By dx dz þ Bz dx dy:
ð82Þ
By using the Frobenius theorem Eq. (82) can be written as
Bx dy dz þ By dx dz þ Bz dx dy ¼ dh1 ^ dh2 :
ð83Þ
The condition for the existence of the function h1, h2 is
X ¼ Pdx þ Qdy þ Rdz
ð84Þ
X ^ dX ¼ PðRy Q z Þdx þ Q ðPz Rx Þdy þ RðQ x Py Þdz ¼ 0;
ð85Þ
dX ¼ B:
ð86Þ
and
where
We notice that the Nambu method is suitable for describing the non-integrable systems. 4. Conclusion Formalization of the magnetic field in Hamiltonian form has advantage of using the mechanical tools as KAM and perturbation theory. For example due to the Hamiltonian character of the magnetic lines the problem of the distributive instabilities can be considered from the point of the canonical perturbation theory. Each equation maybe has many geometric conception. Hamiltonian equations also could be obtain from the geometric ways. In this article we use this method for the magnetic field lines. Also, the geometric formulation of Nambu mechanics is applied to the magnetic field lines and it is found the condition for the existence of Nambu functions for the magnetic field lines.This formalism is suitable for non-integrable systems. In addition of this, a geometrical formulation has many advantages within the dynamical systems, for example, enable us to treat some general issues such as integral invariants and canonical transformations in simple way and also gives one deep insights. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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