classical hamiltonian

7252019 CLASSICAL HAMILTONIAN

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Anomalous State from a Classical Hamiltonian Electric

Dipole in a Magnetic FieldBoris Atenas Luis A del Pino and Sergio Curilef

Departamento de Fiacutesica Universidad Catoacutelica del Norte Avenida Angamos 0610 Antofagasta Chile

Abstract We study the classical behavior of an electric dipole in the presence of a uniform magnetic field Using theLagrangian formulation we obtain the equations of motion whose solutions are represented in terms of Jacobi functions

We also identify two constants of motion namely the energy E and a pseudomomentum C We obtain a relation between theconstants that allows us to suggest the existence of a type of bound states without turning points which are called trappedstates These results are consistent with and complementary to previous results

Keywords General Physics Electric Dipole Lagrangian formulation

In the present day many specialists study the world at the molecular scale Nanotechnology is slowly exploring

molecular rotors and applications of this concept are extensive Using electric fields molecules can change in

orientation andor remain controlled [1 2 3] Molecular-level devices can be obtained from the conversion of energy

into controlled motion nevertheless it is difficult to repeat this process using a mechanical molecular motor although

it is common in biological systems For the time being it is expected that the physical principles at the scale of

a molecular engine can be identified 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 dipolersquos plane of motion

This system has been approached from various perspective by several authors [4 5 6 7] However the trajectory of

the center of mass and the conditions for the existence of trapped states in terms of the constants of motion have not

been fully studied In this article we describe in detail the solution of the equations of motion in the coordinates of the

relative motion and the center of mass which we derive from the Lagrangian formulation of the problem The relation

between the constants of motion which permits the existence of trapped states is established These solutions couldsignificantly impact the future of the applications and construction to molecular motors as they describe the overall

behavior of a dipole from a classical perspective

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 potential A as follows B = nabla times A We assign to the particle 1(2) the charge e1e2 the position

r 1( r 2) the velocity ˙ r 1(˙ r 2) and the mass m1(m2)

L( r 1 r 2 ˙ r 1 ˙ r 2) = 1

2m1

˙ r 12

+ 1

2m2

˙ r 22 minus e1

c A( r 1) middot ˙ r 1minus e2

c A( r 2) middot ˙ r 2minus e1e2

κ | r 2 minus r 1| (1)

where κ is the dielectric constant of the medium in which the motion of charges occurs We define the vector potential A using the symmetric gauge A( r i) = 1

2 B times r i where B is the uniform magnetic field Now we consider the following

change of variables

r = r 1 minus r 2 R = m1 r 1 + m2 r 2

m1 + m2

(2)

where r is the relative position and R is the position of the center of mass Now we consider a rigid dipole composed 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 particles carries charge +e whereas the other carries charge minuse and | r | = a is the fixed

length of the dipole If we substitute finally we obtain the following function

L( R r ˙ R ˙ r ) =

1

2 M

˙ R2

+ 1

2micro ˙ r

2+

e2

κ a +

e

2c

B times R middot ˙ r + B times r middot ˙ R +

(m1minusm2)

M B times r middot ˙ r

983133 (3)



where micro = m1m2 M

is the reduced mass and M = m1 + m2 is the total mass of the dipole The energy of the system can

be obtained by a legendre transformed of langrangian

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

C = P R + q A R + ec Ar (4)

where q = e1 + e2 is the total charge and ec = e1 m2 M minus e2 m1

M is the coupling charge A R = 12c B times R Ar = 1

2c B times r and

P R = M ˙ R + 1

2c B times r Thusthe energy for q = 0 is given by

E = 1

2micro ˙ r

2+

1

2 M

e

c B times r minus C

2minus e2

κ a (5)

Now if we restrict the motion of the particles to a plane perpendicular to eΓ = eC times e B to constitute a complete

set of orthonormal vectors namely a basis In the above basis r and ˙ R are represented in terms of ψ which is the

angle between the vectors r and eΓ In addition we define ρ = r a

ξ = R

a d

dt =ω c

d d τ ω c = eB

Mc (the cyclotron frequency)

˙ ξ = sinψ eΓ + (Γ minuscosψ )eC (6)

The pseudomomentum is also defined as a dimensionless constant as followsΓ

= | C

| M ω ca If we define the Eq(7) in terms of the dimensionless units α = M

micro ε c = 2e2

ω 2c M κ a3 ε = 2E

ω 2c M κ a2 we obtain

ε = 1

α ψ 2 minus2Γ cosψ + 1 +Γ

2minus ε c (7)

By taking the time derivative of the previous equation and dismissing the trivial solution ψ = 0 we obtain

ψ +Ω2 sinψ = 0 with Ω

2 = Γ a (8)

where the time derivative is in terms of the dimensionless time τ We emphasize that Eq(8) coincides with the equation

of motion of a nonlinear pendulum whose general solution is [8]

φ = sgn ψ 0 k Ω[τ

minusτ 0] + snminus1(k 0

| χ )

ψ = 2arcsin[sn(φ (τ )| χ )]sgn(cn(φ (τ )| χ ))

ψ = 2sgn ψ 0 k Ωdn(φ (τ )| χ ) (9)

where χ = 1k

k =

radic ψ 2

0+4Ω2k 2

02Ω

k 0 = sin ψ 0

2 ψ 0 ψ 0 are the initial angular velocity and the initial orientation of the

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

sgn( x) =

983163 1 x ge 0

minus1 x lt 0 (10)

Furthermore we must take into account some additional definitions these are the Jacobi functions [9]

sn( x

|k ) = sin(am( x

|k ))

cn( x|k ) = cos(am( x|k ))

dn( x|k ) =991770

1minusk 2sn2( x|k ) (11)

and am( x|k ) is the inverse of an incomplete elliptic function [9] of the first kind with

x =

int am( x|k )

0

d θ radic 1minusk 2 sin2 θ

(12)

The set of equations (9) is valid for any value of the parameter ξ and the value of this parameter classifies the motion

of the dipole into two possible states if ξ le 1 the dipole has sufficient energy to rotate and if ξ gt 1 the dipole



oscillates around equilibrium For the latter case it is suitable from the numerical point of view to reformulate the set

of equations (9) as a function of the parameter k which takes values in accordance with 0 le k le 1

φ = sgn ψ 0 Ω[τ minus τ 0] + snminus1(k 0k |k )

ψ = 2arcsin[k sn(φ |k )]

ψ = 2sgn ψ 0 k Ωcn(φ |k ) (13)

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

T =

983163 2 χ K ( χ )Ω χ le 1

4 χ K (k )Ω χ gt 1 (14)

where K ( x) is the complete elliptic integral of the first kind

The law of motion that satisfies the position of the center of mass can be found by integrating the Eq(6) with the aid

of Eqs (9) and (13) The solution of the Eq(6) for χ gt 1 can be written as follows

ξ1 = ξ10

minus2k

sgn ψ 0

Ω

[cn(φ

|k )

minuscn(φ 0

|k )]

ξ2 = ξ20 + sgn ψ 0Ω

[(Γ minus1)(φ minusφ 0) + 2φ minusφ 0 + E (am(φ 0|k ))minus E (am(φ |k ))] (15)

where the quantity with subindex zero represents initial conditions

Equations (15) represent the general laws of motion for the position of the center of mass regardless whether the dipole

possesses enough energy to rotate namely for any value of χ Trapped states exhibit the property of having a mean velocity equal to zero[4 7] For the variable of the center of mass

Eqs (15) mean

ξ1(φ prime)minusξ10 = 0 ξ2(φ prime)minusξ20 = 0 (16)

where φ prime = 2sgn ψ 0 k ( χ ) +φ 0 The first equation of (16) is immediately satisfied because of the periodicity of Jacobi

functions The second is satisfied only if

Γ = 1minus2K ( χ )minus E ( χ )

χ 2K ( χ ) χ le 1 Γ = 1minus2

K (k )minus E (k )

K (k ) χ gt 1 (17)

where E ( x) is the complete elliptic integral of the second kind The condition for the existence of trapped states is

included in Eq(17) As concluded previously in [4] trapped states can only exist if Γ le 1 which is the same result as

in Eq(17)

Case 1 χ le 1 represents rotating dipoles Eq (17) has nonnegative solutions if ξ = 0 Γ = 0 Eq (16) represents a

trapped motion In the spatial reference system centered on the initial position of the center of mass the previous

equation is a curve that forms a sort of cardioid

Case 2 χ gt 1 the equation (17) may possess positive solutions if 0 le k lt k max k max asymp 09 By analyzing the equation

(17) in terms of k sim 0 and preserving the terms of the order k 2 Γ sim 1minus k 2 and Eqs(15) can be converted in

ξ1 = ξ10minus2sgn ψ

0Ω

k [cos(φ )minuscos(φ 0)]

ξ2 = ξ20minus sgn ψ 02Ω

k 2[sin(2φ )minus sin(2φ 0)] (18)

The aproximation for k rarr 0 that leads to Eq (18) defines a Lissajous figure with δ = π 2

centered in a suitable system

of reference However if we repeat the calculation numerically for higher values for k surprisingly we obtain a nearly

identical figure As stated above the orbit for all values of the parameter k are very similar In order to show a graphical

example for functions obtained from the Eq(18) we define the following normalized cordinates for the center of mass

ξ1 N = ξ1minusξ10

maxξ1minusξ10 ξ2 N =

ξ2minusξ20

maxξ2minusξ20

(19)



FIGURE 1 The normalized path of the center of mass (of the dipole) for the initial conditions given by ψ 0 = 0 Orbits for two

values of k are compared The path for small k (namely k = 001) is represented by a solid line and the case k = 085 is representedby red points It is apparent that the two curves nearly coincide

In summary the present study supports and generalizes previous analyzes discussed in the literature which can be

considered as particular cases of the present analysis It is interesting to note that the system under certain conditions

can move against the sense of the pseudomomentum As expected the motion in the pseudomomentum sense is

possible too Certainly the present study of the behavior of the electric dipole in presence of a magnetic field is

classical The behavior strongly depends on the initial conditions If we slightly modify the initial conditions for the

system we no longer obtain this special family of states We think this is a relevant contribution that we expect to

follow studying by several perspectives As said before for the time being it is expected that some physical principles

at the scale of a molecular engine can be identified by applying rotor modeled as electric dipoles in external magnetic

fields

We acknowledge partial financial support from CONICYT-UCN PSD-065 and Fondos UCN BA acknowledg-ments the financial support from CONICYT Beca Magister Nacional 2014

REFERENCES

1 V V Volchkov M N Khimich M Ya Melnikov and B M UzhinovA Fluorescence Study of the Excited State Dynamicsof Boron Dipyrrin Molecular Rotors Hihg Energy Chemistry 47(2013)224-229

2 Y Yoshida Y Shimizu T YajimaG Maruta S Takeda Y Nakano T HiramatsuH Kageyama H Yamochi and G SaitoMolecular Rotors of Coronene in Charge-Transfer Solids Chemistry-A European Journal 19 (2013) 12313-12324

3 N Ogiwara T Yanagibashia Y Hikichia M Nishikawaa J Kamiyaa K Wadab Development of a turbo-molecular pumpwith a magnetic shield function Vacuum 98 (2013) 1821

4 Troncoso P Curilef S Bound and trapped states of an electric dipole in a magnetic field European Journal of Physics 27(2006) 1315-1322

5 Curilef S and Claro F Dynamics of two interacting particles in a magnetic field in two dimensions American Journal of Physics 65 (1997) 244-2506 Escobar- Ruiz M A Turbiner AV Two charges on a plane in magnetic field special trajectories Journal of Mathematical

Physics 54 (2013)0229017 Pursey D L Sveshnikov N A Shirokov A M Electric dipole in a magnetic field Bound states without classical turning

points Theoretical and Mathematical Physics 117 (1998) 1262-12738 Karlheinz O A comprehensive analytical solution of the nonlinear pendulum European Journal of Physics 32 (2011)

479-4909 P FByrd and M D Friedmann Handbook of Elliptical Integrals for Engineers and Scientists (New York Springer 1971)

M Abramowitz and I A Stegun Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables(New York Dover 1972) H Hancock rsquoElliptic Integralsrsquo (New York Dover 1958)

10 Atenas B Del Pino LA Curilef S Classical States of an Electric Dipole in an External Magnetic Field Complete solutionfor the center of mass and trapped states Annals of physics in press(2014)



where micro = m1m2 M

is the reduced mass and M = m1 + m2 is the total mass of the dipole The energy of the system can

be obtained by a legendre transformed of langrangian

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

C = P R + q A R + ec Ar (4)

where q = e1 + e2 is the total charge and ec = e1 m2 M minus e2 m1

M is the coupling charge A R = 12c B times R Ar = 1

2c B times r and

P R = M ˙ R + 1

2c B times r Thusthe energy for q = 0 is given by

E = 1

2micro ˙ r

2+

1

2 M

e

c B times r minus C

2minus e2

κ a (5)

Now if we restrict the motion of the particles to a plane perpendicular to eΓ = eC times e B to constitute a complete

set of orthonormal vectors namely a basis In the above basis r and ˙ R are represented in terms of ψ which is the

angle between the vectors r and eΓ In addition we define ρ = r a

ξ = R

a d

dt =ω c

d d τ ω c = eB

Mc (the cyclotron frequency)

˙ ξ = sinψ eΓ + (Γ minuscosψ )eC (6)

The pseudomomentum is also defined as a dimensionless constant as followsΓ

= | C

| M ω ca If we define the Eq(7) in terms of the dimensionless units α = M

micro ε c = 2e2

ω 2c M κ a3 ε = 2E

ω 2c M κ a2 we obtain

ε = 1

α ψ 2 minus2Γ cosψ + 1 +Γ

2minus ε c (7)

By taking the time derivative of the previous equation and dismissing the trivial solution ψ = 0 we obtain

ψ +Ω2 sinψ = 0 with Ω

2 = Γ a (8)

where the time derivative is in terms of the dimensionless time τ We emphasize that Eq(8) coincides with the equation

of motion of a nonlinear pendulum whose general solution is [8]

φ = sgn ψ 0 k Ω[τ

minusτ 0] + snminus1(k 0

| χ )

ψ = 2arcsin[sn(φ (τ )| χ )]sgn(cn(φ (τ )| χ ))

ψ = 2sgn ψ 0 k Ωdn(φ (τ )| χ ) (9)

where χ = 1k

k =

radic ψ 2

0+4Ω2k 2

02Ω

k 0 = sin ψ 0

2 ψ 0 ψ 0 are the initial angular velocity and the initial orientation of the

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

sgn( x) =

983163 1 x ge 0

minus1 x lt 0 (10)

Furthermore we must take into account some additional definitions these are the Jacobi functions [9]

sn( x

|k ) = sin(am( x

|k ))

cn( x|k ) = cos(am( x|k ))

dn( x|k ) =991770

1minusk 2sn2( x|k ) (11)

and am( x|k ) is the inverse of an incomplete elliptic function [9] of the first kind with

x =

int am( x|k )

0

d θ radic 1minusk 2 sin2 θ

(12)

The set of equations (9) is valid for any value of the parameter ξ and the value of this parameter classifies the motion

of the dipole into two possible states if ξ le 1 the dipole has sufficient energy to rotate and if ξ gt 1 the dipole







ψ = 2sgn ψ 0 k Ωcn(φ |k ) (13)


T =

983163 2 χ K ( χ )Ω χ le 1

4 χ K (k )Ω χ gt 1 (14)




ξ1 = ξ10

minus2k

sgn ψ 0

Ω

[cn(φ

|k )

minuscn(φ 0

|k )]

ξ2 = ξ20 + sgn ψ 0Ω





Eqs (15) mean






K (k )minus E (k )

K (k ) χ gt 1 (17)



in Eq(17)







0Ω











ξ2minusξ20

maxξ2minusξ20

(19)













fields


REFERENCES

















ψ = 2sgn ψ 0 k Ωcn(φ |k ) (13)


T =

983163 2 χ K ( χ )Ω χ le 1

4 χ K (k )Ω χ gt 1 (14)




ξ1 = ξ10

minus2k

sgn ψ 0

Ω

[cn(φ

|k )

minuscn(φ 0

|k )]

ξ2 = ξ20 + sgn ψ 0Ω





Eqs (15) mean






K (k )minus E (k )

K (k ) χ gt 1 (17)



in Eq(17)







0Ω











ξ2minusξ20

maxξ2minusξ20

(19)













fields


REFERENCES























fields


REFERENCES











classical hamiltonian

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