Bateman Oscillators: Caldirola-Kanai and Null Lagrangians and Gauge Functions
aa r X i v : . [ phy s i c s . c l a ss - ph ] N ov Bateman Oscillators: Caldirola-Kanai Lagrangiansand Gauge Functions
L. C. Vestal and Z. E. Musielak
Department of Physics, The University of Texas at Arlington, Arlington, TX 76019,USAE-mail: [email protected]
Abstract.
The Lagrange formalism is developed for Bateman oscillators, whichinclude both damped and amplified systems, and a novel method to derive theCaldirola-Kanai and null Lagrangians is presented. For the null Lagrangians,corresponding gauge functions are obtained. It is shown that the gauge functions can beused to convert the undriven Bateman oscillators into the driven ones. Implications ofthe obtained results on quantization of damped dynamical systems are also discussed.
1. Introduction
The Bateman model consists of two uncoupled oscillators, damped or time-forwardand amplified or time-reversed [1], which are called here the Bateman oscillators. Theequations of motion for these oscillators are derived from one Lagrangian that is knownas the Bateman Lagrangian [1,2] and used in studies of the damped harmonic oscillatorand its quantization [3,4]. General solutions of the equations of motion for the Batemanmodel are well-known and given in terms of elementary functions [5].We solve the inverse problem of the calculus of variations [6] for the Batemanoscillators and develop a novel method to derive the standard and null Lagrangians.The standard Lagrangians (SLs) contain the square of the first order derivative of thedependent variable (the kinetic energy like term) and the square of dependent variable(the potential energy like term), and among the derived SLs, the Caldirola-Kanai (CK)Lagrangian is obtained [7,8] and its validity to describe the Bateman oscillators isdiscussed. The null (or trivial) Lagrangians (NLs), when substituted into the Euler-Lagrange (EL) equation, make this equation vanish identically, which means that noequation of motion is obtained. The NLs are also required to be expressed as the totalderivative of a scalar function [9-13], which is called a gauge function [14,15].Our objective is to derive the SLs, NLs and their gauge functions (GFs) for theBateman oscillators and verify them against the three Helmholtz conditions [16,2], whichguarantee the existence of Lagrangians for the conservative systems. One obtained SLis the CK Lagrangian, which is well-known and all derived NLs and GFs are new. TheCK Lagrangian is modified by taking into account the GFs and it is shown that this ateman Oscillators: Caldirola-Kanai Lagrangians and Gauge Functions
2. Lagrangians and gauge functions for Bateman oscillators
The Bateman oscillator model combines damped and amplified oscillators [1], calledhere the Bateman oscillators, into one dynamical system. The model is based on theBateman Lagrangian [1-4] given by L B [ ˙ x ( t ) , ˙ y ( t ) , x ( t ) , y ( t )] = m ˙ x ( t ) ˙ y ( t ) + γ x ( t ) ˙ y ( t ) − ˙ x ( t ) y ( t )] − kx ( t ) y ( t ) , (1)where x ( t ) and y ( t ) are coordinate variables and ˙ x ( t ) and ˙ y ( t ) are their derivatives withrespect to time t . In addition, m is mass and γ and k are damping and spring constants,respectively. Using the EL equations for y and x , the resulting equation of motion forthe damped oscillator is m ¨ x ( t ) + γ ˙ x ( t ) + kx ( t ) = 0 , (2)and the equation of motion for the amplified oscillator becomes m ¨ y ( t ) − γ ˙ y ( t ) + ky ( t ) = 0 . (3)The equations are uncoupled but they are related to each other by the transformation[ x ( t ) , y ( t ) , γ ] → [ y ( t ) , x ( t ) , − γ ], which allows replacing Eq. (2) by Eq. (3) and viceversa.Let us define b = γ/m and c = k/m = ω o , where ω o is the characteristic frequencyof the oscillators, and write Eqs (2) and (3) as one equation of motion¨ x ( t ) + b ˙ x ( t ) + ω o x ( t ) = 0 , (4)with the understanding that the damped and amplified oscillators require b > b <
0, respectively, and that the variable x ( t ) becomes y ( t ) for the amplified oscillator.Let ˆ D = d /dt + bd/dt + ω o be a linear operator, then Eq. (4) can be written in thecompact form ˆ Dx ( t ) = 0.In the following, we derive the SLs, NLs and GFs for the Bateman oscillators whoseequations of motion are ˆ Dx ( t ) = 0. ateman Oscillators: Caldirola-Kanai Lagrangians and Gauge Functions The first-derivative term in Eq. (4) can be removed by using the standard transformationof the dependent variable [17]. The transformation is x ( t ) = x ( t ) e − bt/ , (5)where x ( t ) is the transformed dependent variable, and it gives¨ x ( t ) + (cid:18) ω o − b (cid:19) x ( t ) = 0 . (6)Despite the fact that the first derivative term is removed, the coefficient b is still presentin the transformed equation of motion. However, if b = 0, then x ( t ) = x ( t ) and Eq.(6) becomes the equation of motion for a undamped harmonic oscillator [18,19].The Lagrangian for this equation is L s [ ˙ x ( t ) , x ( t )] = 12 (cid:20) ( ˙ x ( t )) − (cid:18) ω o − b (cid:19) x ( t ) (cid:21) . (7)Having obtained L s [ ˙ x ( t ) , x ( t )], we may now convert the variable x ( t ) into x ( t )using the inverse transformation to that given by Eq. (5). The resulting Lagrangian is L [ ˙ x ( t ) , x ( t ) , t ] = L CK [ ˙ x ( t ) , x ( t ) , t ] + L n [ ˙ x ( t ) , x ( t ) , t ] , (8)where L CK [ ˙ x ( t ) , x ( t ) , t ] = 12 h ( ˙ x ( t )) − cx ( t ) i e bt , (9)is the Caldirola-Kanai (CK) Lagrangian [7,8], derived here independently; the validityof this Lagrangian for the Bateman oscillators is discussed in Section 4. The methodgives the following null Lagrangian L n [ ˙ x ( t ) , x ( t ) , t ] = X i =1 L ni [ ˙ x ( t ) , x ( t ) , t ] , (10)where the partial null Lagrangians are L n [ ˙ x ( t ) , x ( t ) , t ] = (cid:18) C + 12 b (cid:19) (cid:20) ˙ x ( t ) + 12 bx ( t ) (cid:21) x ( t ) e bt , (11) L n [ ˙ x ( t ) , x ( t ) , t ] = C (cid:20)(cid:18) ˙ x ( t ) + 12 bx ( t ) (cid:19) t + x ( t ) (cid:21) e bt/ , (12)and L n [ ˙ x ( t ) , x ( t ) , t ] = C (cid:20) ˙ x ( t ) + 12 bx ( t ) (cid:21) e bt/ + C , (13)where C , C , C and C are constants to be determined. These are new null Lagrangiansfor the Bateman oscillators. The fact that L n [ ˙ x ( t ) , x ( t ) , t ] and its partial Lagrangiansare the NLs can be shown by defining ˆ EL to be the differential operator of the ELequation and verifying that ˆ EL ( L n ) = 0 as well as ˆ EL ( L ni ) = 0. It must be also notedthat b = 0 reduces L n [ ˙ x ( t ) , x ( t ) , t ] to the previously obtained null Lagrangian [15]. ateman Oscillators: Caldirola-Kanai Lagrangians and Gauge Functions For each partial null Lagrangian, its corresponding partial gauge function can beobtained and the results are φ n [ x ( t ) , t ] = 12 (cid:18) C + 12 b (cid:19) x ( t ) e bt , (14) φ n [ x ( t ) , t ] = C x ( t ) te bt/ , (15)and φ n [ x ( t ) , t ] = C x ( t ) e bt/ + C t . (16)These partial gauge functions can be added together to form the gauge function φ n [ x ( t ) , t ] = X i =1 φ ni [ x ( t ) , t ] . (17)The derived gauge function and partial gauge functions reduce to those previouslyobtained [15] when b = 0 is assumed.To construct the general gauge function, we consider the partial gauge functions(see Eqs. 14, 15 and 16) and replace the constants by functions of the independentvariable; it is required that these functions are continuous and differentiable. Then, thegeneral partial gauge functions can be written as φ gn [ x ( t ) , t ] = 12 (cid:20) f ( t ) + 12 b (cid:21) x ( t ) e bt , (18) φ gn [ x ( t ) , t ] = f ( t ) x ( t ) te bt/ , (19)and φ gn [ x ( t ) , t ] = f ( t ) x ( t ) e bt/ + f ( t ) t . (20)The general gauge function is obtained by adding these partial gauge functions φ gn [ x ( t ) , t ] = X i =1 φ gni [ x ( t ) , t ] . (21)The results given by Eqs. (18) through (21) are new general gauge functions for theBateman oscillators and they generalize those previously obtained for Newton’s law ofinertia [20] and for linear undamped oscillators [21] for which b = 0.Using the general gauge function φ gn [ x ( t ) , t ], the resulting general null Lagrangiancan be calculated and the result is L gn [ ˙ x ( t ) , x ( t ) , t ] = X i =1 L gni [ ˙ x ( t ) , x ( t ) , t ] , (22)where the partial null Lagrangians are given by L gn [ ˙ x ( t ) , x ( t ) , t ] = (cid:20) f ( t ) + 12 b (cid:21) (cid:20) ˙ x ( t ) + 12 bx ( t ) (cid:21) x ( t ) e bt + 12 ˙ f ( t ) x ( t ) e bt , (23) ateman Oscillators: Caldirola-Kanai Lagrangians and Gauge Functions L gn [ ˙ x ( t ) , x ( t ) , t ] = h f ( t ) ˙ x ( t ) + ˙ f ( t ) x ( t ) i te bt/ + (cid:18) bt (cid:19) f ( t ) x ( t ) e bt/ , (24)and L gn [ ˙ x ( t ) , x ( t ) , t ] = h f ( t ) ˙ x ( t ) + ˙ f ( t ) x ( t ) i e bt/ + 12 bf ( t ) x ( t ) e bt/ + h ˙ f ( t ) t + f ( t ) i . (25)The general null Lagrangian L gn [ ˙ x ( t ) , x ( t ) , t ] and its partial null Lagrangians reduce tothose previously found [20,21] when b = 0. These Lagrangians depend on four functions,which can be constrainted by the initial conditions if they are specified.Since the null Lagrangians do not affect the derivation of the equations of motion,their existence is not restricted by the Helmholtz conditions [6,16]. However, theconditions seem to imply that the Caldirola-Kanai Lagrangian cannot be constructedfor the Bateman oscillators. In the following, we consider and discuss this problem indetail. The Caldirola-Kanai Lagrangian, L CK [ ˙ x ( t ) , x ( t ) , t ], when substituted into the ELequation, yields [ ˆ Dx ( t )] e bt = 0, which is consitstent with all Helmholtz conditionsthat are valid for conservative systems [16]. However, with e bt = 0, the resultingˆ Dx ( t ) = 0 does obey the first and second Helmholtz conditions but fails to satisfythe third condition. This shows that after the division by e bt the equation of motionfails to satisfy the third condition [22]; see further discussion below Eq. (29).Since the CK Lagrangian depends explicitly on time, the energy function [18,19]must be calculated. Let E CK [ ˙ x ( t ) , x ( t ) , t ] be the energy function for L CK [ ˙ x ( t ) , x ( t ) , t ]given by E CK [ ˙ x ( t ) , x ( t ) , t ] = ˙ x ( t ) p c ( t ) − L CK [ ˙ x ( t ) , x ( t ) , t ] , (26)where the canonical momentum p c ( t ) is p ( t ) = ∂L CK [ ˙ x ( t ) , x ( t ) , t ] ∂ ˙ x ( t ) = ˙ x ( t ) e bt , (27)and is different than the linear momentum p ( t ) = ˙ x ( t ).Then, the energy function E CK [ ˙ x ( t ) , x ( t ) , t ] can be written as E CK [ ˙ x ( t ) , x ( t ) , t ] = E tot [ ˙ x ( t ) , x ( t )] e bt . (28)where the total energy is E tot [ ˙ x ( t ) , x ( t )] = 12 h ( ˙ x ( t )) + ω o x ( t ) i . (29)According to Eqs. (28) and (29), the energy function depends explicitly on time butthe total energy does not. Since the well known solutions for x ( t ) [18] show that ateman Oscillators: Caldirola-Kanai Lagrangians and Gauge Functions e − bt/ for the damped (amplified)oscillators, and since E CK [ ˙ x ( t ) , x ( t ) , t ] depends on squares of x ( t ) and ˙ x ( t ), the resulting e − bt and e bt cancel each other, which means that E CK [ ˙ x ( t ) , x ( t ) , t ] = const, as expected[18,19], and that E tot [ ˙ x ( t ) , x ( t )] decreases (increases) exponentially in time for thedamped (amplified) oscillators.Despite the fact that E tot [ ˙ x ( t ) , x ( t )] = const, E CK [ ˙ x ( t ) , x ( t ) , t ] = E tot [ ˙ x ( t ) , x ( t )] e bt = const, the existence of this constant energy function guarantees that the equation ofmotion [ ˆ Dx ( t )] e bt = 0 resulting from the CK Lagrangian satisfies the third Helmholtzcondition but [ ˆ Dx ( t )] = 0 does not, as it is already pointed out at the begining of thissection.Having obtained E CK [ ˙ x ( t ) , x ( t ) , t ], we use dE CK [ ˙ x ( t ) , x ( t ) , t ] dt = − ∂L CK [ ˙ x ( t ) , x ( t ) , t ] ∂t , (30)and find the following equation of motion[ ˆ Dx ( t )] ˙ x ( t ) e bt = 0 . (31)Since ˙ x ( t ) e bt = 0, Eq. (31) becomes ˆ Dx ( t ) = 0, which is the equation of motion for theBateman oscillators (see Eq. 4).The validity of the CK Lagrangian for the Bateman oscillators has been questionedin the literautre [23] based on the previous work [24]. The main conclusion of thisresearch was that the CK Lagrangian does not describe the Bateman oscillators butinstead a different oscillatory system with its mass increasing ( b >
0) or decreasing( b <
0) exponentially in time. However, more recent work [25] demonstrated that thisconclusion is not correct and that the CK Lagrangian is valid for the Bateman oscillators.The results presented in this paper are consistent with those given in [25] as wedemonstrated that the energy function (see Eq. 26) that depends on the canonicalmomentum p c ( t ) is constant in time, and that the total energy (see Eq. 29) thatdepends on the linear momentum p ( t ) is not constant in time. This shows that the totalenergy and the linear momentum decrease (increase) in time for the damped (amplified)Bateman oscillators, which is consistent with the physical picture of these dynamicalsystems. The increase (decrease) of the canonical momentum in time does not contradictthis picture but instead establishes the validity of the CK Lagrangian for the Batemanoscillators [25].
3. From undriven to driven Bateman oscillators
Having demonstrated the validity of the CK Lagrangian for the Bateman oscillators, wenow investigate effects of the gauge functions on the CK Lagrangian. Let us combinethe CK Lagrangian, given by Eq. (9), and the general null Lagrangian, given byEqs. (22) through (25), together to obtain the following Lagrangian L [ ˙ x ( t ) , x ( t ) , t ] = ateman Oscillators: Caldirola-Kanai Lagrangians and Gauge Functions L CK [ ˙ x ( t ) , x ( t ) , t ] + L gn [ ˙ x ( t ) , x ( t ) , t ]. By using this new Lagrangian, we find the resultingenergy function to be E [ ˙ x ( t ) , x ( t ) , t ] = 12 h ( ˙ x ( t )) + (cid:16) ω o − ¯ ω o ( t ) (cid:17) x ( t ) i e bt − F ( t ) x ( t ) e bt/ − G ( t ) , (32)where ¯ ω o ( t ) = (cid:20)(cid:18) f ( t ) + 12 b (cid:19) b + ˙ f ( t ) (cid:21) , (33) F ( t ) = (cid:20)(cid:18) bt (cid:19) f ( t ) + ˙ f ( t ) t + ˙ f ( t ) + 12 bf ( t ) (cid:21) , (34)and G ( t ) = ˙ f ( t ) t + f ( t ) . (35)We make the frequency shift ¯ ω o time-independent by taking f ( t ) = C andobtaining ¯ ω o = ( C + b/ b . We also define the total energy E tot [ ˙ x ( t ) , x ( t )] = 12 h ( ˙ x ( t )) + ω s x ( t ) i , (36)where ω s = ω o − ¯ ω o = const. It must be noted that E tot [ ˙ x ( t ) , x ( t )] is not constantbecause x ( t ) and ˙ x ( t ) decrease exponentially in time if b > b < E [ ˙ x ( t ) , x ( t ) , t ] = h E tot [ ˙ x ( t ) , x ( t )] − F ( t ) x ( t ) e − bt/ − G ( t ) e − bt i e bt . (37)Since F ( t ) and G ( t ) depend on the functions f ( t ), f ( t ) and f ( t ), which are arbitrary,they can be chosen so that the difference between E tot [ ˙ x ( t ) , x ( t )] and the two energyterms F ( t ) x ( t ) e − bt/ and G ( t ) e − bt approaches zero as t → ∞ .It must be also noted that in the special case of b = 0, Eq. (37) reduces to Eq. (40)derived in [20]. However, if b = 0 and f ( t ) = C , f ( t ) = C and f ( t ) = C , then thetotal energy function is equal to that obtained in [20]. Despite the fact that the partial null Lagrangians were used in deriving the total energyfunctions, these null Lagrangians were derived from the partial gauge functions givenby Eqs. (18) through (20). Based on definitions given by Eqs. (33) through (35) , it canbe seen that the partial gauge function φ gn [ x ( t ) , t ] contributes only to ¯ ω o ( t ), and that F ( t ) is determined by φ gn [ x ( t ) , t ] and also by φ gn [ x ( t ) , t ], which contributes partiallyto both F ( t ) and G ( t ).The extra energies F ( t ) x ( t ) and G ( t ) in E [ ˙ x ( t ) , x ( t ) , t ] can be now addedto L CK [ ˙ x ( t ) , x ( t ) , t ] given by Eq. (9), so that the modified CK Lagrangian L CK,mod [ ˙ x ( t ) , x ( t ) , t ] becomes L CK,mod [ ˙ x ( t ) , x ( t ) , t ] = 12 h ( ˙ x ( t )) − ω s x ( t ) i e bt ateman Oscillators: Caldirola-Kanai Lagrangians and Gauge Functions − F ( t ) x ( t ) e bt/ − G ( t ) , (38)which is the well-known Lagrangian for forced and damped oscillators [18,19]. Ourresults demonstrate that the gauge function is responsible for introducing two extraenergy terms to the original CK Lagrangian.Using L CK,mod [ ˙ x ( t ) , x ( t ) , t ], the following equation of motion is obtained[¨ x ( t ) + b ˙ x ( t ) + ( ω o − ¯ ω o ( t )) x ( t )] e bt = F ( t ) e bt/ , (39)which can also be written as¨ x ( t ) + b ˙ x ( t ) + ω s x ( t ) = F ( t ) e − bt/ . (40)An interesting result is that the solutions to the homogeneous equation of Eq. (40)depend on e − bt/ , which is the exponential factor as in the forcing function; see belowfor the full contemplemantary and particular solutions.The definition of F ( t ) shows that it is determined by the gauge function φ gn [ x ( t ) , t ],which contributes through its function f ( t ) and its derivative, and also partially by φ gn [ x ( t ) , t ], which contributes through its function f ( t ) and its derivative. Moreover, f ( t ) and f ( t ) can be any functions of t as long as they are differentiable. Moreconstraints on these functions can be imposed after invariance of the action is considered[20] or the initial conditions are specified.Let us define β = b/ x ( t ) + 2 β ˙ x ( t ) + ω s x ( t ) = F ( t ) e − βt . (41)Since β = const and ω s = const, the solution to the homogeneous part of Eq. (40) iswell-known [18,19] and given by x h ( t ) = x o e ( − β + iω ) t , (42)where x o is an integration constant and ω = q ω s − β is the natural frequency ofoscillations.To find the particular solution x p ( t ), the force function F ( t ) must be specified. Letus take F ( t ) = F o e ( i Ω) t , (43)where F o and Ω are the constant amplitude and frequency of the driving force,respectively. Then, we seek the particular solution in the following form x p ( t ) = Γ e ( − β + i Ω) t , (44)with Γ is a constant to be determined. Substituting Eq. (44) into Eq. (41) and usingEq. (43), we obtainΓ = F o ω − Ω . (45)Thus, the solution to Eq. (40) is x ( t ) = x o e ( − β + iω ) t + F o ω − Ω e ( − β + i Ω) t . (46) ateman Oscillators: Caldirola-Kanai Lagrangians and Gauge Functions x ( t ) and that found in textbooks ofClassical Mechanics [18,24] is that both x h ( t ) and x p ( t ) decay exponentially in time,which results in a different Γ that shows a resonance if ω = Ω.To derive the standard solution given in the textbooks [18,24], it is required that F ( t ) = F o e ( β + i Ω) t . (47)This is allowed as F ( t ) is arbitrary as long as it is differentiable. However, F ( t ) isexpressed in terms of the functions f ( t ) and f ( t ) and their derivatives, these twofunctions still remain to be determined once F ( t ) is specified. Since f ( t ) and f ( t )are arbitrary, we may take either f ( t ) = C = const and solve a first-order ordinarydifferential equation (ODE) for f ( t ) or assume that f ( t ) = C = const and solveanother ODE for f ( t ).The presented results demonstrate that the CK Lagrangian modified by the gaugefunctions gives the equations of motion for the driven Bateman oscillators. From aphysical point of view, this means that the derived gauge function can be used tointroduce forces to undriven dynamical systems [21]
4. Conclusions
We developed the Lagrangian formalism for Bateman oscillators and used it to derivethe Caldirola-Kanai and null Lagrangians. For each null Lagrangian the correspondinggauge function was obtained. Our results confirm the previous findings [7,8,25] thatthe CK Lagrangian can be used to derive the equations of motion for the Batemanoscillators. We also considered the Helmholtz conditions that guarantee the existenceof Lagrangians for given ODEs, and demonstrated that the energy function obtainedfrom the CK Lagrangian is the sufficient condition for the resulting equation of motionto satisfy the conditions.The results obtained in this paper contradict the previous claims [23,24] that theCK Lagrangian is not valid for the Bateman oscillators but instead it represents adifferent dynamical system in which mass increases or decreases exponentially in time.As a result of this contradiction, derivations of the Kanai-Caldirola propagator in the deBroglie-Bohm theory [26], which are based on the CK Lagrangian with its mass beingtime-dependent [27] must be taken with caution.Our main result is that the CK Lagrangian modified by the presence of extraenergy terms generated by the partial gauge functions can be used to describe thedriven Bateman oscillators. The modified CK Lagrangian given by Eq. (38) and theresulting equation of motion for the driven Bateman oscillators (see Eq. 40) clearly showthat the Lagrangian formulation is possible, and that the undriven Bateman oscillatorscan be converted into the driven ones if the gauge functions are taken into account.In the previous attemps to quantize the Bateman oscillators [4] using the Pais-Uhlenbeck [28] and Feshbach-Tikochinsky [29] methods, the Bateman Lagrangian givenby Eq. (1) was used. Our results imply that the driven Bateman oscillators can also be ateman Oscillators: Caldirola-Kanai Lagrangians and Gauge Functions
5. References5. References