aa r X i v : . [ m a t h - ph ] F e b Relativistic Ermakov-Milne-Pinney Systems and First Integrals
Fernando Haas
Physics Institute, Federal University of Rio Grande do Sul,Av. Bento Gon¸calves 9500, 91501-970 Porto Alegre, RS, Brazil
Abstract
The Ermakov-Milne-Pinney equation is ubiquitous in many areas of physics that have an ex-plicit time-dependence, including quantum systems with time dependent Hamiltonian, cosmology,time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physicalinterpretation of the Ermakov-Lewis invariant is applied as a guiding principle for the derivationof the special relativistic analog of the Ermakov-Milne-Pinney equation and associated first inte-gral. The special relativistic extension of the Ray-Reid system and invariant is obtained. Generalproperties of the relativistic Ermakov-Milne-Pinney are analyzed. The conservative case of therelativistic Ermakov-Milne-Pinney equation is described in terms of a pseudo-potential, reducingthe problem to an effective Newtonian form. The non-relativistic limit is considered as well. A rel-ativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalizedErmakov-Milne-Pinney equation has additional nonlinearities, due to the relativistic effects.
PACS numbers: 0.30.Hq; 03.30.+p; 05.45.-a; 45.20.-d . INTRODUCTION The Ermakov-Milne-Pinney (EMP) equation [1–3]¨ x + κ ( t ) x = C/x , (1)where C is a real constant usually taken as positive, is an ubiquitous nonlinear non-au-tonomous ordinary differential equation with many applications, in particular in problemsrelated to the time-dependent harmonic oscillator or in connection with exact solutions of theone-dimensional time-independent Schr¨odinger equation. In more generality, applications ofthe EMP equation appear in cosmological models [4–6], Bose-Einstein condensates [7, 8],photonic lattices [9], accelerator dynamics [10, 11], gravitational wave propagation [12],higher order spin models [13], quantum plasmas [14], limit cycles [15], dynamical symmetries[16], magneto-gasdynamics [17], time-dependent non-Hermitian quantum system [18, 19],supersymmetric systems [20], noncommutative quantum mechanics [21], etc. Historical notescan be found e.g. in [22, 23]. We note that the name of the EMP equation is not yeta consensus in the literature. For instance, sometimes it can be referred just as Pinneyequation.Diverse generalizations of the EMP equation have been proposed, as for instance allowing C ( t ) to have a dependence on time [8, 23], the inclusion of dissipation [24, 25], unbalancedsystems of EMP equations with different frequency functions [26], modified nonlinearities[27], stochastic differential equations with additive noise [28]. Following the generalizationtrend, it would be relevant to extend the EMP equation into the special relativity domain,with potential applications for the dynamics of charged particles in high energy densityfields [29]. To date only few results weakly related to special relativistic EMP equationsare available, in connection with the Dirac equation [30] and the relativistic isotropic two-dimensional time-dependent harmonic oscillator [31]. Naturally, the relativistic extension isnot straightforward due to the intrinsic extra nonlinearity imposed by the presence of theLorentz factor. Moreover, the intimate connection between the EMP equation and the time-dependent harmonic oscillator raises the question of what defines the relativistic harmonicoscillator. In this regard, the relativistic motion immediately induces anharmonicities evenin the case of a quadratic potential with a frequency-amplitude dependence [32, 33]. Here2e follow the approach of many authors [34–43], adopting the spinless Salpeter equation[44] with a quadratic external potential as our definition of relativistic harmonic oscillator.The choice is due to its simplicity, where just the Newtonian kinetic energy is replaced byits special relativistic counterpart. This version of the relativistic harmonic oscillator modelhas been recently experimentally probed [45].The present work proposes a systematic approach towards the relativistic EMP equationand beyond, using as a simple guiding principle the Eliezer and Gray [46] physical inter-pretation of the Ermakov-Lewis associated first integral for the isotropic two-dimensionaltime-dependent harmonic oscillator. The reason is that the Eliezer and Gray method basedon the conservation of the angular momentum of an auxiliary planar motion provides thestandard physical reasoning behind the conservation of the Ermakov invariant [47, 48]. More-over, the extension of the Ray-Reid Ermakov systems [47] to the relativistic domain will bealso obtained, in a certain sense to be explained. In brief, a relativistic extension of the cele-brated EMP equation and extensions is proposed (which obviously has no direct relation tospecial solutions of Klein-Gordon, Dirac or similar relativistic partial differential equations).This work is organized as follows. In Section II, we review the Eliezer and Gray interpre-tation. In Section III, we consider the relativistic isotropic two-dimensional time-dependentharmonic oscillator and follow the Eliezer and Gray method, in order to identify a relativisticEMP equation together with the associated conservation law. From the structure of the dy-namical equations, the appropriate relativistic Ray-Reid system will be also derived, as wellas with the corresponding first integral. Section IV provides an alternative derivation basedon a dynamical rescaling of time parameter. Sections V and VI deal with basic properties ofthe relativistic EMP equation, mainly in the autonomous case. Section VII is dedicated toa nonlinear superposition law relating the solutions of the relativistic EMP system. Finally,Section VIII is reserved to the conclusions and extra remarks. II. THE ELIEZER AND GRAY PHYSICAL INTERPRETATION
We briefly reproduce the Eliezer and Gray physical interpretation [46] of the Ermakov-Lewis invariant, which will serve us as a guidance for a relativistic generalization of the EMP3ystem. Consider the Lagrangian L = 12 ( ˙ x + ˙ y ) − V ( x, y, t ) , V ( x, y, t ) = κ ( t )( x + y )2 (2)the corresponding auxiliary plane motion¨ r + κ ( t ) r = 0 , (3)where r has the Cartesian components ( x, y ), and the EMP equation¨ ρ + κ ( t ) ρ = J ρ , (4)where J is a real constant. In terms of polar coordinates ( ρ, θ ), where ρ = | r | , x = ρ cos θ, y = ρ sin θ , the equations of motion become¨ ρ − ρ ˙ θ + κ ρ = 0 , (5)1 ρ ddt ( ρ ˙ θ ) = 0 . (6)Equation (6) implies the constancy of the angular momentum J = ρ ˙ θ = const . (7)assuming for simplicity an unit mass. Combining Eqs. (5) and (7) we derive Eq. (4).Considering the equation for the x − component of the auxiliary motion together with Eq.(4), the constancy of the Ermakov-Lewis invariant I given by I = 12 (cid:20) ( ρ ˙ x − x ˙ ρ ) + J x ρ (cid:21) (8)is directly verified, dI/dt = 0 along trajectories. Expressing in terms of polar coordinates,one has I = 12 ( J sin θ + J cos θ ) = J . (9)Therefore, the invariance of I is equivalent to the invariance of the angular momentum ofthe auxiliary plane motion. 4or reference, it is worth to consider the Ray-Reid (RR) generalization [47] of the EMPsystem and invariant, namely ¨ x + κ x = f ( y/x ) yx , (10)¨ y + κ y = g ( x/y ) xy , (11)where f, g are arbitrary functions of the indicated arguments. The RR first integral for Eqs.(10) and (11) is I RR = 12 ( x ˙ y − y ˙ x ) + Z y/x f ( s ) ds + Z x/y g ( s ) ds . (12)One can directly verify that dI RR /dt = 0 along trajectories. III. A RELATIVISTIC ERMAKOV-MILNE-PINNEY SYSTEM
Proceeding in strict analogy with the NR case, consider the equations of motion for the2D relativistic unit rest mass time-dependent harmonic oscillator, which can be derived[49, 50] from the Lagrangian L = − c γ − V ( x, y, t ) , V ( x, y, t ) = κ ( t )( x + y )2 , (13)where c is the speed of light and γ = [1 − ( ˙ x + ˙ y ) /c ] − / . The Euler-Lagrange equationsare (cid:18) − ˙ y c (cid:19) ¨ x + ˙ x ˙ yc ¨ y = − κ xγ , (14) (cid:18) − ˙ x c (cid:19) ¨ y + ˙ x ˙ yc ¨ x = − κ yγ , (15)which can be disentangled as ¨ x + κ γ xγ x = κ γ ˙ x ˙ yc y , (16)¨ y + κ γ yγ y = κ γ ˙ x ˙ yc x , (17)5here γ x = (1 − ˙ x /c ) − / , γ y = (1 − ˙ y /c ) − / . The equations of motion are in agreementwith [31]. Similarly to the NR case, employing cylindrical coordinates x = ρ cos θ, y = ρ sin θ ,we have the conserved angular momentum expressed as J = ∂L∂ ˙ θ = γρ ˙ θ . (18)Moreover, we get the Lorentz factor expressible in terms of ρ, ˙ ρ as γ = (cid:18) J /c ρ − ˙ ρ /c (cid:19) / . (19)The NR formal limit c → ∞ yields γ = 1 also from Eq. (19). Using the angular momentumto eliminate the angular velocity, we obtain¨ x + κ γ (cid:18) x − ρ ˙ ρ ˙ xc (cid:19) = 0 , (20)¨ ρ + κ γ (cid:18) − ˙ ρ c (cid:19) ρ = J γ ρ . (21)From the identity γ ( ρ ˙ x − x ˙ ρ ) = J sin θ , it becomes self-evident that the quantity I R = 12 (cid:20) γ ( ρ ˙ x − x ˙ ρ ) + J x ρ (cid:21) (22)is a first integral, since I R = 12 ( J sin θ + J cos θ ) = J , (23)in complete analogy with the Ermakov-Lewis invariant for the NR problem. It can bealso directly verified that dI R /dt = 0 along the trajectories of the system (20)-(21). Theinvariant has the same physical interpretation of the NR Ermakov invariant in terms of theconservation of the angular motion of the associated auxiliary 2D motion. In this context,it is justified to interpret Eqs. (20) and (21) as the (special) relativistic EMP system.Equation (21) is a relativistic EMP equation (REMP), and the first integral in Eq. (22)is the relativistic Ermakov-Lewis invariant of the problem. Obviously, in the formal limit c → ∞ one recovers the NR case. Understanding the Lorentz factor in the sense of Eq.(19), the relativistic EMP system is a pair of nonlinear second-order ordinary differentialequations for x, ρ . Unlike the NR case, the equation for x is not uncoupled.6o recapitulate, we have just used the standard Lagrangian for the special relativistic two-dimensional motion [51–54], in the particular case where the particle is under the influenceof a force linear in position (a quadratic potential) as seen in a fixed local inertial framewith respect to which the motion is observed [49, 50].In passing, we write the Hamiltonian H associated with (13), H = c (cid:18) p x + p y c (cid:19) / + V ( x, y, t ) , p x = γ ˙ x , p y = γ ˙ y , (24)which is not a constant of motion in the non-conservative case where κ ( t ) is a time-dependentfunction. Even in the autonomous case where d H /dt = 0, the nature of H and the invariant I R in Eq. (22) is different: the former is energy-like, while the latter is angular momentum-like.From the structure of Eqs. (20) and (21) and after some trial and error, it is possible toidentify a relativistic Ray-Reid (RRR) system, namely¨ x + κ γ (cid:18) x − ρ ˙ ρ ˙ xc (cid:19) = (cid:18) − ˙ x c (cid:19) f ( y/x ) γ yx − ˙ x ˙ yγ c g ( x/y ) xy , (25)¨ y + κ γ (cid:18) y − ρ ˙ ρ ˙ yc (cid:19) = − ˙ x ˙ yγ c f ( y/x ) yx + (cid:18) − ˙ y c (cid:19) g ( x/y ) γ xy , (26)where ρ ˙ ρ = x ˙ x + y ˙ y and the Lorentz factor is understood as a function of ˙ x, ˙ y . The invariantfor Eqs. (25) and (26) is I RRR = γ x ˙ y − y ˙ x ) + Z y/x f ( s ) ds + Z x/y g ( s ) ds . (27)It can be verified that dI RRR /dt = 0 along trajectories. It is apparent that Eqs. (25), (26)and (27) define a RRR system and its invariant, showing a complete symmetry betweenthe x and y variables and recovering the RR system and invariant in the formal NR limit c → ∞ , as shown by comparison with Eqs. (10)-(12). A derivation provided by a dynamicalrescaling of time will be described in the next Section.Although our treatment has as motivation the relativistic time-dependent harmonic os-cillator, it is clear that the invariants shown in Eqs. (22) and (27) do not depend on κ .Therefore, one is authorized to allow for more general functional dependencies of κ , e.g. κ = κ ( t, x, y, ˙ x, ˙ y, . . . ) in Eqs. (25) and (26), maintaining the constancy of the RRR invari-7nt. Similar remarks apply to the NR case [55–57]. Notice that Eqs. (25) and (26) do nothave a Lagrangian structure in general. IV. DERIVATION FROM A DYNAMICAL RESCALING OF TIME
Start from the dynamical system x ′′ + ω x = f ( y/x ) yx , y ′′ + ω y = g ( x/y ) xy , (28)where a prime denotes derivative with respect to the independent variable τ , or x ′ = dx/dτ, y ′ = dy/dτ , and where ω can be anything in the same spirit of the last Section.In the same way as for the RR system, it is obvious that (28) possess the invariant I = 12 ( xy ′ − yx ′ ) + Z y/x f ( s ) ds + Z x/y g ( s ) ds , dIdτ = 0 . (29)If we now move to a new independent variable t defined by dtdτ = γ , γ = (cid:18) − ˙ x + ˙ y c (cid:19) − / , ˙ x = dxdt , ˙ y = dydt , (30)then it can be checked after some algebra that the RRR system (25) and (26) is recovered,with κ = ω /γ , and that the invariant (29) transforms into (27). The procedure gives amore transparent derivation of the RRR system from a dynamical rescaling of time startingfrom the RR system.It is worthwhile to have a discussion about Lorentz invariance. In the same sense as theNewtonian harmonic oscillator is obviously not Galilean invariant, the relativistic harmonicoscillator is not Lorentz invariant. In this respect we remember that the symmetry group ofthe NR Ermakov system is the SL (2 , ℜ ) group [58], which has no relation with the Galileangroup. Moreover notice that the force equation dp i /dt = F i where p i are the componentsof the relativistic momentum is relativistic only “in a certain sense”, quoting the wordsfrom Golsdtein’s book [49], since time has been keep as entirely distinct from the spatialcoordinates. However, a fully covariant interpretation is possible, regarding (13) as theLagrangian for a charged particle under a scalar electrostatic potential which is a quadraticfunction of the spatial coordinates, in the instant inertial frame with respect the motion is8bserved. In this context, a fully covariant formalism could be easily performed followingthe standard approach [49–54] starting with the Lagrangian L = − mc /γ + q A · r − qφ (the well-known notation is employed) in the two-spatial dimensions case, with a scalarpotential φ ∼ ( x + y ) together with A = 0. This is the Lagrangian (13) assuming unitrest mass without loss of generality. In addition, the relativistic harmonic oscillator andREMP equations reproduce the Newtonian systems in the limit u/c →
0, where u is ameasure of the maximal velocity of the problem. For instance if A is the amplitude of themotion under a linear force F = − κ x , the relativistic effects become negligible provided κA /c →
0. The case of a very strong external field acting on a charged particle [29] is asuitable system to probe the relativistic effects. In this context since the non-relativisticErmakov system has found applications in accelerator physics [11], the relativistic versionhas potential applications for charged particle motions under high-intensity external fields.Notice that the derivation of the equations (20) and (21) containing the REMP equation,together with the relativistic Ermakov-Lewis invariant (22), follows a very different routecompared to the RRR system (25)-(26) and associated invariant (27). To obtain the REMPequation, the starting point was the relativistic Lagrangian for the 2D time-dependent har-monic oscillator, together with polar coordinates, exactly the same as the procedure for theNR EMP system but then with the special relativistic Lagrangian. On the other hand,the RRR system can be considered “relativistic” only in the ad hoc sense that it reducesto the NR Ray-Reid system in the limit c → ∞ where c is some reference velocity, sinceits derivation does not starts from any physical relativistic Lagrangian. In the same con-text, the use of a dynamical rescaling of time clearly has not a relativistic spirit. First ofall, the γ factor in Eq. (30) is defined in terms of the particle velocity, not the relativevelocity between two inertial frames. Second, it is obviously not a Lorentz transformationsince the space variables are kept the same. Third, we had not the objective of setting asort of symmetry transformation for the equations (28). Rather, the goal was the use ofa non-invariance transformation towards the derivation of the RRR system. Although atthe moment there is the lacking of a a physical interpretation of the RRR system, which asdiscussed is relativistic only in a formal sense, we think it is worthwhile to present it sinceit has a striking analogy with the traditional RR system, being a non-trivial pair of couplednonlinear second-order ordinary differential equations possessing an exact invariant.9 . THE J = 0 CASE
For a vanishing angular momentum, one has ˙ θ = 0 so that it can be chosen θ = 0, withoutloss of generality. In this case, y = 0 and Eq. (20) becomes ddt ( γ ˙ x ) = − κ x , γ = (cid:18) − ˙ x c (cid:19) − / , (31)which is [59] the equation for an one-dimensional (1D) relativistic time-dependent harmonicoscillator.In the case of a constant frequency κ , obviously the energy is conserved. In terms ofrescaled variables ¯ x = κx/c, ¯ t = κt, ¯ v = d ¯ x/d ¯ t , the corresponding first integral H D ≥ H D = (1 − ¯ v ) − / + ¯ x , (32)with phase-space contour plots shown in Fig. 1. Clearly, only bounded trajectories areadmissible, as expected. The return points are located at ¯ x = ±√ H D − / . In rescaledcoordinates, the NR limit H D ≃ v + ¯ x ≃ H D − H D , the return points are also larger, implyingmore relativistic effects and increasing anharmonicity, as seen in Fig. 1. FIG. 1: Phase-space contour plots of the energy first integral (32) for the 1D conservative relativisticharmonic oscillator described by Eq. (31) with constant κ , for ¯ x = κx/c, ¯ v = ˙ x/c and differentvalues of H D , as indicated.
10t is instructive to rewrite the conservation law in a Newtonian form,¯ v V D (¯ x ) = 0 , (33)where the pseudo-potential V D (¯ x ) is defined by V D (¯ x ) = −
12 + 12( H D − ¯ x / , (34)shown in Fig. 2. In the NR limit, the variable ¯ x is limited to small values, so that V D =cte . + ¯ x / (2 H D ) + . . . with H D ≃
1, while larger values of H D correspond to enhancedrelativistic effects and anharmonicity. The quantity V D provides a kind of transfer of thenonlinearity from the kinetic energy term to an anharmonic (pseudo) potential term, butseemingly has not a physical interpretation. FIG. 2: Pseudo-potential V D (¯ x ) from Eq. (34) and different values of H D , as indicated. The conserved energy can be used for the quadrature of the motion in terms of ellipticfunctions. For this purpose, we set H D = 1 + A , − A < ¯ x < A , (35)11here A is the amplitude of the motion, and define¯ x = A sin φ , φ = φ (¯ t ) . (36)After some algebra, Eq. (33) is rewritten as (cid:18) dφd ¯ t (cid:19) = 1 + A cos φ (1 + A cos φ ) , (37)which can be immediately integrated yielding √ A E (cid:18) φ, A √ A (cid:19) − √ A F (cid:18) φ, A √ A (cid:19) = ¯ t − ¯ t , (38)where ¯ t is an integration constant and F, E are incomplete elliptic integrals of the first andsecond kind, respectively defined [60] according to F ( φ, k ) = Z φ dφ ′ (1 − k sin φ ′ ) / , E ( φ, k ) = Z φ (1 − k sin φ ′ ) / dφ ′ . (39)Using MATHEMATICA it is easy to produce a series solution, φ (¯ t ) = ¯ t − ¯ t − A
32 (2(¯ t − ¯ t ) + sin[2(¯ t − ¯ t )]) + O ( A ) , (40)recovering the Newtonian result in the limit of small amplitude (which is also the non-relativistic limit). Similarly the period T for which φ = 2 π is T = 2 π (cid:18) A (cid:19) + O ( A ) . (41)Approximate periodic solutions for the 1D conservative relativistic harmonic oscillator canalso be found in [61]. VI. RELATIVISTIC CONSERVATIVE ERMAKOV-MILNE-PINNEY EQUATION
The REMP defined in Eq. (21) with J = 0 does not have collapsing ( ρ →
0) solutionsdue to the inverse cubic term. For instance, suppose κ = cte . and the rescaling ¯ ρ = κρ/c, ¯ t =12 t, ¯ v = d ¯ ρ/d ¯ t, ¯ J = κJ/c , so that d ¯ vd ¯ t + 1 γ (1 − ¯ v ) ¯ ρ = ¯ J γ ¯ ρ . (42)The energy first integral is H = γ + ¯ ρ ≥ , γ = (cid:18) J / ¯ ρ − ¯ v (cid:19) / . (43)It is immediate to conclude that 0 < ¯ ρ < √ H .Rewriting in a Newtonian form yields¯ v / V ( ¯ ρ ) = 0 , (44)with a pseudo-potential defined by V ( ¯ ρ ) = −
12 + 1 + ¯ J / ¯ ρ H − ¯ ρ / , (45)shown in Fig. 3. It is possible to show that V ( ¯ ρ ∗ ) <
0, where 0 < ¯ ρ ∗ < H is the equilibriumpoint of V , viz. ¯ ρ ∗ = 12 (cid:16) − J + p J + 16 ¯ J H (cid:17) / . (46) FIG. 3: Pseudo-potential V from Eq. (45). The return points correspond to V = 0, similarly to Fig. 2, with stable oscillations13round ¯ ρ ∗ . Elementary algebra shows that the condition for periodic motions ( V ( ¯ ρ ∗ ) < J < F ( H ) = 427 (cid:0) − H + H + (3 + H ) / (cid:1) , (47)or, sufficiently small angular momentum. In the NR limit one has ¯ J < ( H − disre-garding O ( H − terms, so that the periodicity condition can be shown to be automat-ically satisfied. The characteristic function F is shown in Fig. 4. A numerical investiga-tion shows that the periodicity condition is always meet, for meaningful initial conditions0 < ¯ ρ (0) < √ H, − < ¯ v (0) <
1, as expected.
FIG. 4: Characteristic function F ( H ) from Eq. (47) for H ≥ The quadrature of Eq. (44) can be made in terms of elliptic functions, after determiningthe return points corresponding to the potential well from Eq. (45). However, the result isexceedingly complicated.
VII. NONLINEAR SUPERPOSITION LAW
Suppose ρ = ρ ( t ) a particular solution of the REMP equation (21). Introducing the newvariables Q = xρ , T = Z dtγρ , (48)where γ = γ ( t ) is given by Eq. (19) converts the relativistic Ermakov-Lewis invariant (22)into I R = 12 (cid:18) dQdT (cid:19) + J Q , (49)14ormally the same as the energy first integral for a 1D conservative NR harmonic oscillator.A quadrature yields Q = √ I R | J | sin( J T + δ ) , (50)or x = ρ sin (cid:18) J Z dtγρ + δ (cid:19) , (51)since I R = J /
2, adopting
J > δ is a constant phase. The nonlinear superpo-sition law (51) generalizes the Newtonian result [55] to the relativistic context. In concreteapplications, typically the particular solution ρ should be numerically found. VIII. CONCLUSION
In this work, considerable progress was achieved, in the generalization of Ermakov systemstowards the special relativity domain. The Eliezer and Gray physical interpretation of theErmakov-Lewis invariant, was used as a guide for the derivation of the relativistic analog ofthe EMP equation, together with the corresponding first integral for the relativistic planartime-dependent harmonic oscillator. General aspects of the relativistic EMP equation havebeen addressed, and a nonlinear superposition law was derived. In spite of the successfulresults, it is still possible to derive other classes of relativistic Ermakov systems, not arisingfrom the correspondence with the Eliezer and Gray physical interpretation. For instance,symmetry principles can be a guiding principle, although the SL (2 , ℜ ) group structure ofnon-relativistic Ermakov systems obviously tends to be broken in the relativistic domain. Inthe same footing, to carry to the relativistic domain the linearization of standard Ermakovsystems would be a probably unfeasible task. In future works it is worthwhile to see how onecan actually solve the new relativistic Ermakov systems with an explicit time-dependence.For this purpose the search for quasi-exact solutions [62] can be a fruitful approach.15 cknowledgments This research was funded by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecno-l´ogico (CNPq). [1] Ermakov, V.P. Differentzial’nyya uravneniya vtorago poryadka. Usloviya integriruemosti vkonechnom vide.
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