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Annals of Physics 350 (2014) 605614

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Annals of Physics

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Classical states of an electric dipole in anexternal magnetic field: Complete solution forthe center of mass and trapped states

Boris Atenas, Luis A. del Pino, Sergio CurilefDepartamento de Fsica, Universidad Catlica del Norte, Avenida Angamos 0610, Antofagasta, Chile

h i g h l i g h t s

Bound states without turning points. Lagrangian Formulation for an electric dipole in a magnetic field. Motion of the center of mass and trapped states. Constants of motion: pseudomomentum and energy.

a r t i c l e i n f o

Article history:

Received 29 April 2014Accepted 11 August 2014Available online 20 August 2014

Keywords:

General physicsElectric dipoleLagrangian formulation

a b s t r a c t

We study the classical behavior of an electric dipole in the presenceof a uniform magnetic field. Using the Lagrangian formulation, weobtain the equations of motion, whose solutions are representedin terms of Jacobi functions. We also identify two constants ofmotion, namely, the energyEandapseudomomentumC. We obtaina relation between the constants that allows us to suggest theexistence of a type of bound states without turning points, whichare called trapped states. These results are consistent with andcomplementary to previous results.

2014 Elsevier Inc. All rights reserved.

1. Introduction

In the present day, many specialists study the world at the molecular scale. Nanotechnology isslowly exploring molecular rotors, and applications of this concept are extensive. Using electric fields,

Corresponding author.E-mail addresses:[email protected],[email protected](S. Curilef).http://dx.doi.org/10.1016/j.aop.2014.08.0070003-4916/ 2014 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.aop.2014.08.007http://www.elsevier.com/locate/aophttp://www.elsevier.com/locate/aopmailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.aop.2014.08.007http://dx.doi.org/10.1016/j.aop.2014.08.007mailto:[email protected]:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.aop.2014.08.007&domain=pdfhttp://www.elsevier.com/locate/aophttp://www.elsevier.com/locate/aophttp://dx.doi.org/10.1016/j.aop.2014.08.007


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606 B. Atenas et al. / Annals of Physics 350 (2014) 605614

molecules can change in orientation and/or remain controlled[13]. Molecular-level devices can beobtained from the conversion of energy into controlled motion; nevertheless, it is difficult to repeatthis process using a mechanical molecular motor, although it is common in biological systems. Forthe time being, it is expected that the physical principles at the scale of a molecular engine can beidentified by applying rotor dynamics in two dimensions. These rotors are modeled as electric dipoles

in electric or magnetic fields.The primary goal of the present work is to describe the motion of a classical electric dipole in the

presence of an external magnetic field, perpendicular to the dipoles plane of motion. This systemhas been approached from various perspective by several authors[47]. However, the trajectory ofthe center of mass and the conditions for the existence oftrapped statesin terms of the constants ofmotion have not been fully studied. In this article, we describe in detail the solution of the equationsof motion in the coordinates of the relative motion and the center of mass, which we derive from theLagrangian formulation of the problem. The relation between the constants of motion, which permitsthe existence of trapped states, is established.

As previously discussed, a model of rigid and non-rigid dipoles is considered, constraining themotion of the center of mass to a direction that is perpendicular to the magnetic field [7]. The motion

of the relative coordinate into the plane is defined by the direction of the magnetic field and a directionperpendicular to the motion of the center of mass. It is possible to show that for certain values of thecharacteristic parameters defined in the problem, there is a functional relation between two constantsof motion that allows the existence of trapped states [4]; this relation has not yet been analyticallyestablished. In other words, an interval of values is found for the constant of motion where solutionsare possible and its trend of these solutions is well defined for certain limiting values. These states arecalled classical bound states embedded in a continuum. The quantum analogue is also discussed [7].

In addition, equations and constants of motion are found for the model of an electric dipolein an external magnetic field, and a preliminary discussion of the existence of trapped states isintroduced [4]. In addition, a model of two interacting particles is discussed [6], and special trajectoriesare found in this model for several initial conditions of the velocity, direction, charges and values of the

magnetic field. The distance between particles may vary, but the conditions constrain the motion to aplane perpendicular to the field and to a fixed distance between particles. Furthermore, the classicaldynamics of two interacting particles becomes an interesting problem where the challenge is to findsolutions that are fully analytical[47].

These solutions could significantly impact the future of the applications and construction tomolecular motors, as they describe the overall behavior of a dipole from a classical perspective. Thispaper is organized as follows: In Section 2 we present the theoretical basis of the system, deriving theequations and constants of motions. In Section3the solution of the equation of motion is obtainedfor the center of mass coordinates. In Section4we address the conditions that lead to trapped states,as mentioned above. Finally, in Section5,we offer some concluding remarks.

2. Basic definitions and equations of motion

In the present model [4], we consider two charges in the presence of a uniform magnetic field. The

magnetic field is obtained from a vector potentialA, as follows:B= A. We assign to the particle1(2)the chargee1(e2), the positionr1 (r2), the velocityr1(r2)and the massm1(m2). The Lagrangianformulation leads to the following expression:

L(r1, r2;r1,r2)= 12

m1r21+1

2m2r22

e1

cA(r1)r1 e2

cA(r2)r2 e1e2

|r2r1| , (1)

where is the dielectric constant of the medium in which the motion of charges occurs. We definethe vector potentialAusing the symmetric gauge as follows:

A(ri)= 12Bri, fori=1, 2, (2)


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B. Atenas et al. / Annals of Physics 350 (2014) 605614 607

whereBis the uniform magnetic field. Now, we consider the following change of variables:r=r2r1, (3)

R

=

m1r1+m2r2m1+m2

, (4)

wherer is the relative position andR is the position of the center of mass. Now, if we substituteEqs.(2)(4)into Eq.(1),we obtain the following function:

L(R, r;R,r)= 12

MR2 + 1

2r2 e1e2

|r|e1+e2

2cBRR

e2m1e1m22cM

B

Rr+rR (e1m

22+e2m21)2cM2

Brr, (5)where =m1m2/Mis the reduced mass andM=m1+ m2is the total mass of the dipole. Hereafter,we consider arigid dipolecomposed of an internal coupling that holds the two charges together and

ensures that the Coulomb interaction between the charges is constant. Then, one of the particlescarries charge+e, whereas the other carries chargee, the masses are equal m1= m2 anda is thefixed length of the dipole. Thus, the Lagrangian is

L(R, r;R,r)= 12

MR2 + 1

2r2 + e

2

a+ e

2c

BRr+BrR

. (6)

The energy of the system

E(R, r;R,r)=PRR+prrL(R, r;R,r), (7)where

pr

= L

ris the relative conjugate momentum andPR

= L

R, the conjugate momentum for the

center of mass. It is additionally found, for the energy, to be:

E(R, r;R,r)= 12

MR2 + 1

2r2 e

2

a. (8)

The other constant of motion that appears from the analysis is called pseudomomentum[4,6,8],

C=PR+qAR+ecAr, (9)whereq= e1+e2 is the total charge andec= e1 m2Me2 m1M is the coupling charge,AR= 12cBR,

Ar= 12cBrandPR= MR+ e2cBr. Thus, the relation between the conjugate momentum of thecenter of mass and the relative coordinates, forq

=0 is given by

C=PR+ e2cBr (10)

by considering the definition of the conjugate momentum for the center of mass, the constant of

motionCyieldsC= MR+ e

cBr. (11)

By combining Eqs.(8)and(11),we can express the energy in terms of only the relative variable and,

the relation betweenEandC:

E= 12

r2 + 12M

C ecBr2 e2

ka. (12)

Up to this point, this result corresponds to the general motion of the rigid dipole in a constant magneticfield.


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Now, if we restrict the motion of the particles to a plane perpendicular to the magnetic field B, then

thepseudomomentumCbecomes perpendicular to the magnetic field. This choice allows us to defineunit vectors in suitable directions; these are,eC, parallel to the vectorCandeB, parallel to the vectorB. Additionally, we can define a third unit vector

e=eCeB (13)to constitute a complete set of orthonormal vectors, namely, a basis.

The expressions forrandR, in the above basis, are=cos e+sin eC, (14)= sin e+(cos )eC, (15)

where is the angle between the vectorsrande, = ra ,= Ra , ddt= c dd, c= eBMc, is thecyclotron frequency. The pseudomomentum is also defined as a dimensionless constant as follows:

= |

C|/M

ca. If we define Eq.(8)in terms of the dimensionless units introduced before, then

=

d

d

2

+ 1

dd

2

c, (16)

where= M

,c= 2e22cMa

3 ,= 2E2ca2Mand by considering Eqs.(14)and(15),we obtain

= 1

2 2 cos +1+2 c. (17)

By taking the time derivative of the previous equation and dismissing the trivial solution= 0, weobtain

+ 2 sin = 0, with 2 =, (18)where the time derivative is in terms of the dimensionless time . We emphasize that Eq.(18)coin-cides with the equation of motion of a nonlinear pendulum whose general solution is[9]:

=sgn0 k[0] +sn1(k0| ),=2 arcsin[sn(()| )]sgn(cn(()|)), (19)=2sgn0 k dn(()| ),

where

= 1

k

,k

=

20 +42k20

2

,k0=

sin 0

2

,

0, 0 are the initial angular velocity and the initial

orientation of the dipole, respectively, and the sgn(x)function is defined as

sgn(x)=

1 x0,1 x< 0. (20)

Furthermore, we must take into account some additional definitions; these are the Jacobi functions[10]

sn(x|k)=sin(am(x|k)),cn(x|k)=cos(am(x|k)), (21)dn(x

|k)

= (1

k2sn2(x

|k))

and am(x|k)is the inverse of an incomplete elliptic function[10]of the first kind, with

x= am(x|k)

0

d(1k2 sin2())

. (22)


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The set of equations(19)is valid for any value of the parameter and the value of this parameterclassifies the motion of the dipole into two possible states: if 1, the dipole has sufficient energyto rotate and, if >1, the dipole oscillates around equilibrium. For the latter case, it is suitable, fromthe numerical point of view, to reformulate the set of equations(19)as a function of the parameterk,which takes values in accordance with 0

k

1:

=sgn0 [0] +sn1

k0

k

k

,

= 2 arcsin[ksn(|k)], (23)= 2sgn0 kcn(|k).

The angle and the angular velocity are periodic functions with the following period:

T=

2 K()/ 1,4K(1/)/ >1,

(24)

whereK(x)is the complete elliptic integral of the first kind.

3. Motion of the center of mass

The law of motion that satisfies the position of the center of mass can be found by integrating Eq.(15),with the aid of Eqs.(19)and(23),and is written as follows:

1=2 sn(| )cn(| ), (25)2=1+2sn2(| ), (26)

where

1and

2are the components of the velocity of the center of mass in the directions

e

and

e

Crespectively. Eq.(25)is easily integrable, so

d(dn(x| ))dx

= 2sn(x| )cn(x| ),

d1

d= 4sgn

0E0 2

d(dn(| ))d

, (27)

1=104 sgn0

E02[dn(| )dn(0| )],

where10 is the initial condition for 1, and the parameters 0 and E0 are: 0

= sn1(k0

| ) and

E0= 02 +4 2k20, respectively. The solution of Eq.(26)can also be obtained from the primitive ofthe function sn2(x| )[11] as 1

2[xE(am(x| ))], whereE(x| )is an incomplete elliptic function of

the second kind:

2=20+2 sgn0

E02

(1)(0)+ 2

2{0+E(am(0| ))E(am(| ))}

. (28)

Eqs. (27) and (28) represent the general laws of motion for the position of the center of mass regardlesswhether the dipole possesses enough energy to rotate, namely, for any value of . To clarify theformulation, we rephrase Eqs.(27)and(28)for >1 as a function ofkas follows:

1=102 k sgn0

[cn(|k)cn(0|k)], (29)

2=20+ sgn0

[(1)(0)+2{0+E(am(0|k))E(am(|k))}] .


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Consider two sets of initial conditions that correspond to limiting cases of the integrals of motion.

Case 1: (Fixed dipole):0=0, 0=0, k=0The set of equations(29)becomes the set

1=

10,

2=20+(1) (0) , (30)which represents a free motion of the fixed dipole in the direction defined byC. We candistinguish two possibilities depending on the value of . This is, if > 1, the motion is

parallel toC. If < 1, the motion is antiparallel toC, which is the same trajectory observedby Curilef et al.[5]and Troncoso et al. [6]. For a direct derivation of this last sentence, seeAppendix.The resting system corresponds to the marginal case where =1.

Certainly, the marginal case is quite unexpected. Charges are at rest in the relativecoordinate. Therefore, only the Coulomb force is present. By contrast, the center of mass

moves perpendicular to the direction of the constant of motionC. Indeed, such a motion ofthe center of mass is additionally perpendicular to the relative vector between particles, thusboth particles move on straight lines with the same velocity. This fact permits the appearing,by the Lorentz force, of an electric force which cancels out the Coulomb interaction betweencharges. Besides, all vectors involved in this motion are perpendicular among them andcoincide with the definition of the basis given by Eq. (13). Alternatively, we remark thatthe combination of the parameters corresponds to a motion that occurs in the minimum ofthe effective potential that can be derived from Eq. (12).Thus, the system with both chargesconstrained by the fixed distance naturally move, on straight lines with the same velocity,with an energy level that exactly coincides with the minimum of the effective potential.

Case 2: (Rotating dipole): =0, 0= 0eBIn this case, the chosen basis becomes meaningless because the vectorseCande are not

defined. The new basis is 0,eB 0andeB, with0= 0. Thus,k= 02 ,= 0 and

the set of equations(19)become the set

= 02

,

= 0 , (31)= 0,

which represents a rotation with constant angular velocity for the relative variable that isone of the special trajectories identified in [6].

Now, by inserting the set of equations given in Eq. (31) into Eq. (15) and performing the subsequentintegration, we can write:

=0+ 10

(0) . (32)

In the limit 0 0, Eq.(32)corresponds toCase 1. In other words, the center of mass moves freelyin the direction perpendicular to the constant orientation of the dipole.

4. Trapped states

Trapped states exhibit the property of having a mean velocity equal to zero[4,7]. For the variableof the center of mass, Eqs.(27)and(28)mean:

1()10=0,

2()20=0, (33)

=2 sgn0 k( )+0.


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Fig. 1. The pseudomomentum (from Eq.(11))is depicted as a function of from Eq.(34).

The first equation of(33)is immediately satisfied because of the periodicity of Jacobi functions. Thesecond is satisfied only if

=12 K( )E( ) 2K( )

1,

=12 K(k)E(k)K(k)

>1, (34)

whereE(x) is the complete elliptic integral of the second kind. The condition for the existence oftrapped states is included in Eq.(34).As concluded previously in [4], trapped states can only exist if1, which is the same result as in(34).Case 1: 1 represents rotating dipoles Eq. (34) has nonnegative solutions (Fig. 1) if= 0, =0.

Eq.(32)represents a trapped motion. In the spatial reference system centered on the initialposition of the center of mass, the previous equation is a curve that forms a sort of cardioid(seeFig. 2).

Case 2: >1Eq.(34)may possess positive solutions if 0k < kmax,kmax0.9. By analyzing Eq.(34)

in terms ofk 0 and preserving the terms of order k2; 1k2 and Eqs.(29)can beconverted into the following:

1102sgn

0

k(cos cos 0),

220 sgn0

2k2(sin(2)sin(20)). (35)

The approximation fork 0 that leads to Eq.(35)defines a Lissajous figure with= 2

centered in a suitable system of reference. However, if we repeat the calculation numericallyfor higher values ofk, surprisingly, we obtain a nearly identical figure. As stated above, theorbits for all values of the parameterkare very similar.

In order to show a graphical example for the functions obtained from Eq.(35),we definethe followingnormalizedcoordinates for the center of mass

1N= 110Max(110) ,

2N= 220Max(220) , (36)


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Fig. 2. The normalized orbit of the center of mass.K= 0, = 0, which form a cardioid.

Fig. 3. The normalized path of the center of mass(of the dipole) for the initial conditions given by0=0. Orbits for two valuesofkare compared. The path for small k(namely,k=0.01) is represented by a solid line, and the case k=0.85 is representedby red points. It is apparent that the two curves nearly coincide.

where Max(x)means the maximum value of the variablesx. Thus, inFig. 3we depict1Nversus2N, with 0=0.

5. Concluding remarks

In summary, the present study supports and generalizes previous analyzes discussed in theliterature, which can be considered as particular cases of the present analysis.

The Lagrangian function is first explicitly written in terms of particle coordinates, but after a suit-able change of variables we write the Lagrangian in terms of center of mass and relative coordinates.

All subsequent analysis is performed using these coordinates. Thus, the motion of the center of massis essentially negligible and trapped states are only briefly mentioned. Here, the equations of motionare integrated directly and precisely, establishing the conditions for the existence of trapped states.These conditions are in agreement with previous results[4] and establish the relation between thevalues of constants of motionCand Ethat enables the existence of trapped states.


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The integration is performed in terms of special functions, such as Jacobi functions [10] that havebeen previously defined. We show a complete analytical solution for the problem, classifying sep-arately the motion of the center of mass and the relative motion. In one hand, we derive the twocomponents for the trajectory of the motion of the center of mass. Two limiting cases that emphasizeparticular initial conditions are shown, these are specifically the fixed dipole and the rotating dipole.

First, it is interesting to note that the system, under certain conditions, can move against the sense ofthe pseudomomentum. As expected, the motion in the pseudomomentum sense is possible too. In thelatter case, a rotation with constant angular velocity is obtained for the relative variable previouslyidentified in[6].

On the other hand, the relevant observation on the existence of a kind of special states is madethat we have called: trapped states. This is a family of states, which are appointed in the specialmotion of the relative coordinate. The main property of having a mean velocity equal to zero withoutturning points is the fingerprint of this kind of interesting states. Here, we again identify two limitingcases. The first limiting case, a cardioid-like curve is shaped (see Fig. 2) in the spatial reference systemcentered on the initial position of the center of mass. Another limiting and interesting case, whichis graphically illustrated inFig. 3,is the identical behavior exhibited by the normalized paths of the

center of mass for several values ofk. A comparison between the small values ofk and k= 0.85 ispresented inFig. 3,where it can be seen that the curves are very similar.

Certainly, the present study of the behavior of the electric dipole in presence of a magnetic fieldis classical. The behavior strongly depends on the initial conditions. If we slightly modify the initialconditions for the system, we no longer obtain this special family of states. We think this is a relevantcontribution that we expect to follow studying by several perspectives. As said before, for the timebeing, it is expected that some physical principles at the scale of a molecular engine can be identifiedby applying rotor modeled as electric dipoles in external magnetic fields.

Acknowledgments

We acknowledge partial financial support from CONICYT-UCN PSD-065. In addition, one of us (B.A.)acknowledges the financial support from CONICyT/Beca Magster Nacional 2014, Folio 22140054.

Appendix

The constant of motion,C, whose dependence on the conjugate momentum of the center of massand the relative coordinates is given, in Eq.(11),by

C= MR+ ecBr, (37)

whose compound in one of the elements of the basis(13),eC, yieldsCeC= MReC+ e

cBreC, (38)

which can be written in terms of the dimensionless variables, defined in the text, as follows:

=C+eBeeC 1

. (39)

Now, the physical meaning of this quantity imposes

=C+10. (40)Therefore,

1=C 1. (41)Thus,1C 0, the dipole has a motion parallel to the pseudomomentum.


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