Rotary dynamics of the rigid body electric dipole under the radiation reaction
aa r X i v : . [ phy s i c s . c l a ss - ph ] A ug Rotary dynamics of the rigid body electric dipoleunder the radiation reaction
A. DuviryakInstitute for Condensed Matter Physics of NAS of Ukraine,1 Svientsitskii Street, Lviv, UA-79011, UkraineTel.: +380 322 701496, Fax: +380 322 [email protected]
Abstract
Rotation of a permanently polarized rigid body under the radiation reactiontorque is considered. Dynamics of the spinning top is derived from a balance condi-tion of the angular momentum. It leads to the non-integrable nonlinear 2nd-orderequations for angular velocities, and then to the reduced 1st-order Euler equations.The example of an axially symmetric top with the longitudinal dipole is solved ex-actly, with the transverse dipole is analyzed qualitatively and numerically. Physicalsolutions describe the asymptotic power-law slowdown to stop or the exponentialdrift to a residual rotation; this depends on initial conditions and a shape of thetop.Keywords: Radiation reaction, spinning topPACS: 41.60.-m, 45.40.-f, 02.30.Ik
It was reported recently that silica particles of mass about 1 fg and size 100 nm were spinedup in the optical trap to the frequency above 1 GHz [1] . This corresponds to the orbitalquantum number of order ℓ & , i.e., the rotational motion is quite classical. If such aparticle possesses an electric dipole moment then it emits the electromagnetic radiationand receives the reaction torque which slows the particle rotation down. Of course, thisrelativistic effect is very small for the aforementioned particle whose constituents moveno faster than 10 − of light speed. But if the spinning particle is kept free for a long time,its radiative slowdown can be measurable and deserves a theoretical study.The classical dynamics of a single point-like charge is governed by the relativistic Lorentz-Dirac equation or its slow-motion predecessor, the
Abraham-Lorentz equation[3,4]. Both include the radiation reaction terms which depend on 3rd-order derivatives andgive rise to redundant runaway solutions. To get rid of these solutions one uses frequentlythe approximated 2nd-order reduction of the Abraham-Lorentz-Dirac equations in which This record has been improved to 5.2 GHz [2]. § The slowdown of a charged plane rotator can be deduced easily from the energy balancecondition, by means of the Larmor formula [3–5]. Instead, we proceed by methodologicalpurpose from the Abraham-Lorentz equation: m ˙ v = F + 2 q c ¨ v , (2.1)where v and ˙ v ≡ d v / d t are velocity and acceleration of particle of mass m and charge q , and c is the speed of light. The last term in r.-h.s. of eq. (2.1) describes the radiationreaction force; it causes physical effect only in the presence of other forces F .Let F be a reaction of holonomic constraint which admits a circular motion, i.e., turnsthe particle into the plane rotator. Then the equation (2.1) reduces to the form:˙Ω = τ ( ¨Ω − Ω ) , where τ = 2 q mc , (2.2)which henceforth will be referred to as the rotator Lotentz equation. This is the 2nd-orderequation for the angular velocity Ω. Moreover, similarly to the Abraham-Lorentz equation(2.1), equation (2.2) is singularly perturbed, i.e., there is a small time-scale parameter τ in r.-h.s. at the higher-order derivative ¨Ω. Thus this equation possesses redundant set ofsolutions which must be separated out from the physical solutions.Exact solutions of (2.2) are unknown. In Appendix A the equation (2.2) is analyzedqualitatively and numerically, and selection rules for the physical solutions are formulated.Here we use in the r.-h.s. of the equation (2.2) the truncated form of this equation,˙Ω = O ( τ ), and arrive at the approximately reduced 1st-order equation:˙Ω = − τ Ω . (2.3)It admits a standard Cauchy problem and possesses the following solution:2( t ) = Ω p τ Ω t , Ω(0) = Ω . (2.4)The characteristic quantity T = 1 / ( τ Ω ) is a time during which a rotary braking ismost intense. Asymptotically, at t ≫ T , the angular velocity decreases by the power lawΩ ∼ / √ τ t which does not depend on the initial value Ω . In Appendix A this result isobtained by the asymptotical analysis of the rotator Lorentz equation (2.2). A plane rotator is the simplest spinning system. Here we consider a composite particleconsisting of point-like charges q a ( a = 1 , , . . . ) with masses m a located at positions r a .We proceed from the slow-motion balance equation [5, § L = 23 c d × ... d (3.1)for the angular momentum L = P a m a r a × v a , where d = P a q a r a is the dipole momentof the system.The composite particle is considered as a free rigid body, i.e., a top. A rotationalmotion of an arbitrary point r ( t ) of the top can be presented as follows: r ( t ) = O ( t ) ρ ,where O ( t ) ∈ SO(3) is a rotation matrix, and ρ is a constant position of the point in theproper reference frame of the top. We will use the following kinematic relations: v ≡ ˙ r = ˙ O ρ = O ( Ω × ρ ) , ˙ v = O { Ω × ( Ω × ρ ) + ˙ Ω × ρ } , ¨ v = O { Ω × [ Ω × ( Ω × ρ ) + 2 ˙ Ω ρ ] + ˙ Ω × ( Ω × ρ ) + ¨ Ω × ρ } , (3.2)where Ω is the angular velocity vector, dual to the skew matrix Ω ≡ O T ˙ O .Using the relations (3.2) in eq. (3.1) yields the equation of rotary motion of a top: I ˙ Ω + Ω × I Ω = 23 c d × { d × (Ω Ω − ¨ Ω ) + ( d · ˙ Ω ) Ω + 2( d · Ω ) ˙ Ω } ; (3.3)here I = || I ij || ( i, j = 1 , ,
3) is the inertia tensor, d ≡ O T d = P a m a ρ a is a constantdipole moment of the top in the proper reference frame, and Ω ≡ | Ω | .The 1st-order reduction of the equation (3.3) implies the elimination the 2nd-orderderivative in r.-h.s. by means of the truncated equation and its differential consequence:˙ Ω = − I − ( Ω × I Ω ) , ¨ Ω = − I − ( ˙ Ω × I Ω + Ω × I ˙ Ω )= I − { ( Ω × ( Ω × I Ω ) − ( I Ω ) × I − ( Ω × I Ω ) } . (3.4)General explicit form of the reduced equation is cumbersome and omitted here.3 .1 Dynamics of an axially symmetric spinning top with thelongitudinal dipole Here we limit ourself by the case of spinning top with an axially-symmetric inertia ellipsoid(i.e., spheroid) and the longitudinal dipole moment: I ij = I i δ ij , I = I ; d = d = 0 , d ≡ d. (3.5)Then the equation (3.3) splits into the following set: I ˙Ω = ( I − I )Ω Ω + 2 d c n ¨Ω − Ω Ω − ˙Ω Ω − ˙Ω o , (3.6) I ˙Ω = ( I − I )Ω Ω + 2 d c n ¨Ω − Ω Ω + ˙Ω Ω + 2Ω ˙Ω o , (3.7) I ˙Ω = 0 . (3.8)It follows from eq. (3.8) that Ω = const provided I = 0. Otherwise Ω remainsan undetermined function of time. From the physical viewpoint, in the case I → Oρ axis, and a rotation in thisdirection is indefinite.The remaining set of equations (3.6)–(3.7) is not solved exactly. In Appendix B anasymptotic behavior of their solutions at t → ±∞ , I ˙Ω = ( I − I )Ω Ω − d c (cid:26) Ω + Ω + I I Ω (cid:27) Ω , (3.9) I ˙Ω = ( I − I )Ω Ω − d c (cid:26) Ω + Ω + I I Ω (cid:27) Ω . (3.10)If I = 0, the equation (3.8) yields Ω = const. The remaining nonlinear equations(3.9)–(3.10) determine components Ω , Ω which form the vector Ω ⊥ = { Ω , Ω , } . Thenone multiplies the equation (3.9) by Ω , the equation (3.10) by Ω , so their sum yields anintegrable equation for Ω ⊥ ≡ Ω ⊥ . Substituting this integral back into the set (3.9)–(3.10)reduces the latter to a linear set. A final integration yields the solution:Ω = Ω ⊥ cos ˜Ω t, Ω = − Ω ⊥ sin ˜Ω t, (3.11)Ω ⊥ ≡ | Ω ⊥ | = ( I /I ) | Ω | r(cid:16) I Ω I Ω ⊥ (cid:17) exp n d I Ω I c t o − , (3.12)where ˜Ω ≡ (1 − I /I )Ω and Ω ⊥ = Ω ⊥ | t =0 . (Here the choice of a reference frameprovides that Ω | t =0 >
0, Ω | t =0 = 0).It follows from (3.11) that the precession of Ω ⊥ has opposite directions for a prolate( I < I ) and oblate ( I > I ) top, and is absent for a spherical top.In the limit Ω → ⊥ :Ω ⊥ = Ω ⊥ q d Ω ⊥ I c t . (3.13)Besides, this expression is true in the case I → and a direction of the vector Ω ⊥ (but not its length Ω ⊥ ) become indefinite, and eqs. (3.11)become meaningless. 4 Dynamics of an axially symmetric spinning topwith the transverse dipole
The case of the spinning top with the dipole moment perpendicular to a symmetry axis, I ij = I i δ ij , I = I ; d ≡ d, d = d = 0 , (4.1)is more cumbersome than that with parallel moment. The reduced equation of motionsplits into the following ones: I ˙Ω = ( I − I )Ω Ω , (4.2) I ˙Ω = ( I − I )Ω Ω − d c (cid:26) Ω + ( I − I )(2 I − I ) I Ω (cid:27) Ω , (4.3) I ˙Ω = − d c (cid:26) Ω + I − I I (2Ω − Ω ) (cid:27) Ω . (4.4)One finds easily two partial solutions of the set (4.2)-(4.4).1). Ω = Ω = 0. Eqs. (4.2), (4.3) become identities while (4.4) reduces to theequation ˙Ω = − d I c Ω (4.5)which, upon redefinition of parameters, coincides with the flat rotator equation (2.3). Thepower-law solution of the type (2.4) tends to zero, Ω →
0, at t → ∞ .2). Ω = 0. Then Ω = const by (4.2) while (4.4) reduces to the equation˙Ω = − d I c { Ω + Ω } Ω . (4.6)It possesses solution on the type (3.12) with the exponential asymptotics Ω → Ω ∞ = { Ω ∞ , , } , Ω ∞ ∈ R .There are no other points which are fixed for the set (4.2)-(4.4).General exact solution of the set (4.2)-(4.4) is unknown. Farther some qualitativeanalysis is undertaken. For this purpose let us introduce the dimensionless quantities: τ = 2 d I c , τ = tτ , ω = τ Ω , (4.7)and change Cartesian components ω , ω by cylindrical ones ω ⊥ , ϕ : ω = ω ⊥ cos ϕ, ω = ω ⊥ sin ϕ ; (4.8)here the angle ϕ determines a direction of the vector ω ⊥ = { ω , ω , } , and ω ⊥ = | ω ⊥ | .In these terms the equations (4.2)-(4.4) take the form:˙ ω ⊥ = −{ ω ⊥ + (1 + δ + δ ) ω } ω ⊥ sin ϕ, (4.9)˙ ω = − − δ { (1 − δ + 3 δ cos ϕ ) ω ⊥ + ω } ω , (4.10)˙ ϕ = − δω − { ω ⊥ + (1 + δ + δ ) ω } sin 2 ϕ, (4.11)where δ ≡ − I /I , − ≤ δ <
1; (4.12)5 ≷ δ = 0 and the disc at δ = −
1. The case δ = 1 of thin rod has no meaning since thevariables ω , ϕ and the direction of dipole are indefinite.Note some general properties of the equations (4.9)–(4.11).It follows from eq. (4.10) that d ω / d τ S ω T
0, i.e., | ω | → τ → ∞ .It follows from eq. (4.9) that ω ⊥ → τ → ∞ with undeterminedfinal angle ϕ ∞ . Otherwise one may occur ω ⊥ → | ω ∞ | > ϕ →
0. But actualbehavior of the angle ϕ is not evident from the equation (4.11) and needs some analysis. It is noteworthy that the r.-h.s of the set (4.9)-(4.11) is a π -periodic function of ϕ . Aver-aging this set yields the equations for averaged variables u ≡ ¯ ω ⊥ , v ≡ ¯ ω , ¯ ϕ :˙ u = −{ u + (1 + δ + δ ) v } u, (4.13)˙ v = − − δ { (2 + δ ) u + 2 v } v, (4.14)˙¯ ϕ = − δ ¯ ω . (4.15)This closed set of equations possesses the exact solution in a parametric form: u = ξv = Cξ − q | (1 + 2 δ ) ξ + 1 + δ | p , (4.16) τ − τ = 1 − δC (cid:12)(cid:12)(cid:12)(cid:12)Z ξξ d ξ ξ q − { (1 + 2 δ ) ξ + 1 + δ } − p − (cid:12)(cid:12)(cid:12)(cid:12) , (4.17)¯ ϕ − ¯ ϕ = ± δ (1 − δ ) √ C Z ξξ d ξ ξ q − | (1 + 2 δ ) ξ + 1 + δ | − p − , (4.18)where q = 21 + δ , p = δ (1 − δ )(3 + 3 δ + δ )(1 + 2 δ )(1 + δ ) , (4.19)and C > τ → ∞ : − ≤ δ ≤ − /
2) ¯ ω ⊥ ∼ | ¯ ω | ∼ τ − / ; | ¯ ϕ | ∼ τ / ; (4.20) − / < δ <
1) ¯ ω ⊥ ∼ τ − / , | ¯ ω | ∼ τ − δ/ − δ , | ¯ ϕ | ∼ τ − δ − δ ) . (4.21)It is seen from (4.21) that the set (4.13)–(4.15) makes a sense for δ < | ¯ ϕ | is aninfinitely increasing function at τ → ∞ . On the contrary, | ¯ ϕ − ¯ ϕ | is asymptoticallydecreasing function at δ > ϕ itself tends to zero or some finite value ϕ corresponding to a fixed point of the original set of equations (4.9)-(4.12).Thus let us return to the equations (4.9)-(4.11) and analyze them in the neighborhoodof the fixed point Ω ∞ = { Ω ∞ , , } . In terms of dimensionless polar coordinates this6oint is determined by ω ⊥ = ω ∞ ≡ τ | Ω ∞ | , ω = 0, ϕ = kπ with k = 0 , ± , ± , . . . . Thelinearization of (4.5)-(4.9) in a neighborhood of fixed points yields the set of equations˙ ν = 0 , (4.22)˙ ω = − δ − δ ω ∞ ω , (4.23)˙ ψ = − δ ω − ω ∞ ψ (4.24)for the deviations ν = ω ⊥ − ω ∞ , ω = ω − ψ = ϕ − kπ . This set possesses exponentialsolutions ∝ e λτ with the characteristics λ = 0, λ = − δ − δ ω ∞ , λ = − ω ∞ . The root λ = 0 generates the change of the fixed value ω ∞ → ω ∞ + ν which corresponds to anearby fixed point. Fixed points with ω ∞ = 0 are stable (due to the linear approximation)for δ > − / δ ≤ − /
2. Once the system gets into the neighborhood ofa stable fixed point, the latter will be reached necessarily.
The analysis of the original (4.9)–(4.11), the averaged (4.13)–(4.15) and the linearized(4.22)–(4.24) sets of equations shows that there are tree different cases. 1). If 0 ≤ δ < < I ≤ I ), the fixed point Ω ∞ = { Ω ∞ , , } with any Ω ∞ = 0 is stable. Thus, starting from arbitrary Ω = { Ω , Ω , Ω } , after fewrevolutions, the vector Ω tends to Ω ∞ exponentially with the characteristic braking time T = { (3 I /I − τ Ω ∞ } − . It is noteworthy that the asymptotic components arisesΩ | t →∞ ≡ Ω ∞ = 0 even if Ω = 0. 2). For the markedly oblate top of − ≤ δ ≤ − / I ≤ I ≤ I ) this fixed point Ω ∞ = { Ω ∞ , , } with Ω ∞ = 0 is unstable. Thus,by general properties of the set (4.9)–(4.11) and of (4.16)–(4.18), the vector Ω continuesto decrease by the asymptotic power law Ω ∼ / √ τ t and reaches the value Ω = 0 afteran infinite number of revolutions around the axis O ρ . 3). In the case − / < δ < I < I < I ) both scenarios are possible, and the finalstate depends on the starting point. These cases are summarized in figure 1. Numericalintegration of the set (4.9)–(4.11) with initial conditions chosen randomly confirms thispicture. There are shown at the figure 2 two examples of evolution of the top withFigure 1: Asymptotic behavior of an axially symmetric top with the transverse dipolemoment d depending on the aspect ratio I /I of the inertia spheroid.7 otary dynamics of the rigid body electric dipole 13 a). b). Fig. 2
Evolution of the axially symmetric top with the transverse dipole moment for 5, = 1 5. a). π/
30. b). π/ = 1 5 but with different π/
30 and π/
3. The example a) istypical for the prolate top, the b) – for the markedly oblate top.
The equation of a rotary motion of the rigid body with the electric dipole moment(3.3) is derived from the Landau-Lifshitz angular momentum balance condition [5].
Figure 2: Evolution of the axially symmetric top with the transverse dipole moment for δ = − / ω ⊥ = ω = 1 /
5. a). ϕ = π/
30. b). ϕ = π/ δ = − / ω ⊥ = ω = 1 / ϕ = π/
30 and ϕ = π/
3. The example a) is typical for the prolate top, the b) – for themarkedly oblate top.
The equation of a rotary motion of the rigid body with the electric dipole moment (3.3) isderived from the Landau-Lifshitz angular momentum balance condition [5]. This equationof the 2nd order with respect to the angular velocity Ω can be reduced approximately tothe 1st-order Euler-type equation.The specific case of a body with the axially-symmetric inertia ellipsoid is considered.The Euler equations for the spinning top with the longitudinal dipole is axially-symmetricand integrable. The longitudinal component Ω of the angular velocity Ω is conservedwhile the transverse component Ω ⊥ decreases exponentially at t → ∞ with the brakingtime T = 1 / ( τ Ω I /I ), where τ is defined by (4.7). If Ω = 0, then Ω decreases8symptotically (at t ≪ T ) by the power law Ω ∼ / √ τ t independently of the initialvalue of Ω, similarly to the case of a plane rotator.Absence of a symmetry may complicate the dynamics. Here it is considered the exam-ple of an axially-symmetric top but with transverse dipole which breaks the symmetry.In this case the dynamics is not integrable and has been analyzed qualitatively and nu-merically. It turned out that a final state of the top depends notably on its shape. Ifthe spinning top is prolate, i.e., I < I , some residual component of the the angularvelocity Ω along the dipole survives (even if this component was absent initially) whileother components decrease exponentially. The markedly oblate top (i.e., if I > I )stops asymptotically by the power law Ω ∼ / √ τ t . For a weakly oblate top (i.e., if I < I < I ) both scenarios are possible, and the final state depends the initial condi-tions.The most realistic case is the asymmetric top. Its dynamics is more complicated.Preliminary calculations reveal exponential drift to some residual rotation if the top ispolarized along a stable principal axis. Otherwise, the top slows down to a complete stop.Study of this braking in detail would be desirable.Finally, let us consider some numerical estimates relevant to the experiment [1] withthe silica particle mentioned in the Introduction. The inertia moment of a sphericalparticle is I = mR , where m = 1 fg and R = 50 nm. Being ionized once up to theelementary charge q = e = 4 . · − CGSE, the particle acquires the dipole moment d = eR ≈ τ ≈ − s is extremely small, but the charac-teristic braking time for Ω = 2 π GHz is astronomical: T ≈ · s ∼ . · years.Hypothetically, these figures may have relevance to the interstellar dust which consists ofsilicate-graphite grains of size 50-500 nm [9]. Grains can be ionized by cosmic rays [10]and spinned up by circularly polarized radiation from relativistic sources such as a blackhole [11]. But GHz rotation of grains seems unlikely.More realistic is a laboratory spin up (and subsequent slowdown) of artificial nanopar-ticles which may carry a large permanent electric dipole moment. The examples areJanus-like particles [12] or nanocrystals CdSe and CdS which at the size ∼ ∼
300 nm ×
30 nm–elongated nanocrystals are reported to possess the mo-ment & ×
100 nm ×
50 nm–crystal is estimated to have the moment & . · D [15, pp. 387-390]. The characteristic braking time for this last example with the initial Ω = 2 π GHzis T ∼ . Appendix
A. Analysis of the plane rotator Lorentz equation
In terms of dimensionless variables τ = t/τ , ω = τ Ω the nonlinear differential 2nd-orderequation (2.2) becomes free of any parameter:¨ ω − ˙ ω − ω = 0; (A.1)9ere the dot “ ˙ ” denotes differentiation by τ . The equation (A.1) is invariant undertime translations τ → τ + λ , λ ∈ R but the corresponding integral of motion is unknown.The change of variable τ → θ = e τ reduces the equation (A.1) to the form:d ω / d θ = ω /θ , (A.2)which is of the Emden-Fowler type equation d y / d x = x n y m . The pair of indices n = − m = 3 does not correspond to integrable cases of this equation [18].Let us study the asymptotic behavior of solutions [19] of the equations (2.2) or (A.1)at t → ±∞ ,
0. We suppose a power-law or exponential asymptotic behavior of solutions. t → + ∞ ). Substituting the power-law anzatz ω = A τ α [1 + O ( τ − )] with the constants A and α to be found into eq. (A.1) leads to the equality: Aα ( α − τ α − − Aατ α − − A τ α = O ( τ α − ) + O ( τ α − ) + O ( τ α − ) . (A.3)The 1st term in l.-h.s. is negligibly small as to the 2nd term which, in turn, can becanceled by the 3rd term, provided α = − / A = 1 /
2. In dimensional terms thiscan be summarized as follows:˙Ω ∼ − τ Ω ⇒ Ω( t ) = ± √ τ t [1 + O ( t − )] , t → + ∞ . (A.4) t → ). Similarly, one obtains the asymptotics: ω ( τ ) = ± √ τ [1 + O ( τ )].Since the equation (2.2) is invariant under the time translation t → t − t by the arbitrary t , this asymptotics can be presented in dimensional terms as follows:¨Ω ∼ Ω ⇒ Ω( t ) = ± √ t − t [1 + O ( t − t )] , t → t , ∀ t ∈ R . (A.5) t → −∞ ). The equation (2.2) or (A.1) does not admit a power-law asymptotics, but theEmden-Fowler equation (A.2) does: ω ∼ Aθ = A e τ , θ → +0 ⇔ τ → −∞ , where A isan arbitrary real constant. In dimensional terms we have:˙Ω ∼ τ ¨Ω ⇒ Ω( t ) ∼ A exp( t/τ ) /τ , t → −∞ . (A.6)The asymptotics (A.5) describes unlimited self-acceleration of a circulating particleduring a finite time. Remarkably, the small parameter τ drops out from this asymptotics.Thus the corresponding solution must be regarded as obviously nonphysical.The asymptotics (A.6) is obviously non-analytical in τ . It is a segment of non-physicalsolution of eq. (2.2) which, in turn, is similar (at t → −∞ ) to the runaway solution ofthe original Lorentz equation (2.1).The only asymptotics (A.4) is physically meaningful since it correlates completely withthe solution (2.4) of the reduced equation (2.3).More details on solutions of the equation (A.1) can be seen in the phase portrait ofthe rotator. Let us recast for this purpose the 2nd order (with respect to ω ) equation(A.1) into the dynamical system: ˙ ω = ̟, (A.7)˙ ̟ = ̟ + ω . (A.8)10t follows from this the Abel equation for phase trajectories of the system:d ̟ d ω = 1 + ω ̟ . (A.9)By now this equation is not solvable, but the asymptotics (A.4), (A.5), (A.6) suggestcorresponding asymptotics of phase trajectories: τ → + ∞ ) ω ∼ ± (2 τ ) − / → ,̟ ∼ ∓ (2 τ ) − / → (cid:27) ⇒ ̟ ∼ − ω . (A.10) ∀ τ ¯ τ ≡ τ − τ → ω ∼ ±√ / ¯ τ → ±∞ ,̟ ∼ ∓√ / ¯ τ → ∓∞ ) ⇒ ̟ ∼ ∓ ω √ . (A.11) τ → −∞ ) ω ∼ A e τ → ,̟ ∼ A e τ → (cid:27) = ⇒ ̟ ∼ ω. (A.12)The system possesses a fixed point O = ( ω = 0 , ̟ = 0), which is unstable. This followsfrom the behavior in neighborhood of this point of the following Lyapunov function [20]: V ( ω, ̟ ) = ω ( ̟ − ω ) : (A.13) V (0 ,
0) = 0; V (0 ≶ ω ≶ ̟ ) >
0; ˙ V = ̟ + ω > . (A.14)A phase portrait of a charged plane rotator derived by means of the above analysisand a numerical integration of the equation (A.9) is presented in figure 3. It is dividedby four domains by two separatrices AOC and
BOD crossing in the fixed point O . Inthe neighborhood of O the separatrix AOC is described by the asymptotics (A.10) while
BOD by the asymptotics (A.12). Opposite i.e. infinite asymptotics of these separatricesas well as the asymptotics of other phase trejectories are described by eq. (A.11), andare reachable in a finite time. Thus all the phase trajectories are non-physical, except theseparatrix
AOC .It is obvious from figure 3 that the set (curve)
AOC is a repeller, or, following theSpoon’s terminology, a critical manifold [6]. Every solution passing through an arbitrarypoint beyond the curve
AOC goes to infinity in a finite time, so it is non-physical.Every physical solution passes points of the curve
AOC . Thus it satisfies both the2st-order equation (A.1) as well as the 1st-order equation˙ ω = f ( ω ) , (A.15)where the function f ( ω ) determines the curve AOC by the equation ̟ = f ( ω ) and thusmeets the following conditions: f ( ω ) = − ω + f ( ω ) d f ( ω ) / d ω , (A.16) f ( ω ) g ω → − ω . (A.17)In other words, the equation (A.15)-(A.17) is the exact 1st-order reduction of the 2nd-order equation (A.1). Since the equation (A.16) or (A.9) is not integrable, one can use ananalytic approximation for f ( ω ). The approximation f ( ω ) ≈ − ω represents in eq. (A.15)a dimensionless form of the reduced equation (2.2). It is precise in the neighborhood of ω → f ( ω ) ≈ − ω / √ ω with which theequation (A.15) is integrable analytically. 11igure 3: Phase portrait of the charged plane rotator. B. Analysis of the Euler-Lorentz equations for a longitudinaldipole
The set of equations (3.6)–(3.8) is not solved exactly. Here an asymptotic behavior ofsolutions at t → ±∞ , ω ⊥ − ˙ ω ⊥ − ω ⊥ − ω ⊥ ν = 0 , (B.1) ω ⊥ ( ˙ ν − ν + ζ ω ) + 2 ˙ ω ⊥ ν = 0 , (B.2) ζ ˙ ω = 0 , (B.3)where ν ≡ ˙ ϕ + ω , ζ = I /I , and the dot “ ˙ ” denotes the differentiation over τ . τ → + ∞ ). The set (B.1)–(B.2) admits a power-law asymptotics provided ω = 0 and/or ζ = 0. In these cases the equation (B.2) possesses the integral of motion C = ω ⊥ ν e − τ which permits us to eliminate ν from the equation (B.1):¨ ω ⊥ − ˙ ω ⊥ − ω ⊥ − C e τ /ω ⊥ = 0 . (B.4)This equation admits a power-law asymptotics at the only value C = 0. Then it becomesidentical to the equation (A.1) for a plane rotator. Thus we have: ω ⊥ = ± √ τ [1 + O ( τ − )] , (B.5) ϕ = ϕ if ω = 0 , ζ = 0 . (B.6)12t ζ = 0 the solution ϕ = ϕ − R τ d τ ω ( τ ) loses a meaning since ϕ is unobservable.In the case ζ = 0, ω = const = 0 the set (B.1)–(B.2) does not admit a power-lawasymptotics at τ → + ∞ . Thus let us look for the exponential asymptotics. The changeof the variable τ by θ = e τ reduces the equations (B.1)–(B.2) to the form: θ ω ′′⊥ − ω ⊥ − ω ⊥ ν = 0 , (B.7) ω ⊥ ( θν ′ − ν + ζ ω ) + 2 θω ′⊥ ν = 0 , (B.8)where the prime “ ′ ” denotes the differentiation with respect to θ . For asympotices (at θ → ∞ , i.e., τ → ∞ ) for ω ⊥ and ν we use the ansatzes: ω ⊥ = Aθ α [1 + O ( θ − )] , ν = Bθ β [1 + O ( θ − )] , (B.9)where A , α , B , β are real constants to be found. The substitution of these anzatzes intothe set (B.7)–(B.8) leads to the equations: α ( α − − A θ α − B θ β = O ( θ − ) + O ( θ α − ) + O ( θ β − ) , (2 α + β − Bθ β + ζ ω = O ( θ β − ) + O ( θ − ) . They are compatible at the only values β = 0, α <
0, and lead to a 4th-degree equationfor α with two real roots:(2 α − α ( α −
1) = ( ζ ω ) = ⇒ α ± = 12 ± r (cid:16) p ζ ω ) (cid:17) ≷ . (B.10)In the present case the only α − is relevant, and we have: α − ≈ − ( ζ ω ) + 5( ζ ω ) + . . . ; B = ζ ω − α − ≈ ζ ω − ζ ω ) + . . . , (B.11)If a value of the variable ω = τ Ω is small, one can retain only leading terms of theseexpansions, and so obtain from (B.9) the asymptotics: ω ⊥ ∼ Aθ α − ≈ A e − ( ζω ) τ , ϕ − ϕ = − ( ω − B ) τ ≈ − (1 − ζ ) ω τ, (B.12)where A is an arbitrary constant. Non-physical asymptotics for the top and the rotator are similar. If ζ = 0: τ → ω ⊥ ∼ ± √ τ [1 + O ( τ )]; ϕ − ϕ = − (1 − ζ ) ω τ + O ( τ ); (B.13) τ → −∞ ) ω ⊥ ∼ Aθ α + ≈ A e τ , ϕ − ϕ ≈ − (1 + ζ ) ω τ, (B.14)where α + is defined in eq. (B.10). If ζ = 0 expressions for ϕ in eqs. (B.13)–(B.14) losemeaning. Acknowledgements
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