On the analytic representation of Newtonian systems
aa r X i v : . [ phy s i c s . c l a ss - ph ] J un On the analytic representation of Newtonian systems
BENOY TALUKDAR, ∗ SUPRIYA CHATTERJEE, and SEKH GOLAM ALI Department of Physics, Visva-Bharati University, Santiniketan 731235, India Department of Physics, Bidhannagar College, EB-2, Sector-1, Salt Lake, Kolkata 700064, India Department of Physics, Kazi Nazrul University, Asansol 713303, India
We show that the theory of self-adjoint differential equations can be used to provide a satis-factory solution of the inverse variational problem in classical mechanics. A Newtonian equationwhen transformed to the self-adjoint form allows one to find an appropriate Lagrangian represen-tation (direct analytic representation) for it. On the other hand, the same Newtonian equationin conjunction with its adjoint provides a basis to construct a different Lagrangian representation(indirect analytic representation) for the system. We obtain the time-dependent Lagrangian of thedamped Harmonic oscillator from the self-adjoint form of the equation of motion and at the sametime identify the adjoint of the equation with the so called Bateman image equation with a viewto construct a time-independent indirect Lagrangian representation. We provide a number of casestudies to demonstrate the usefulness of the approach derived by us. We also present similar resultsfor a number of nonlinear differential equations by using an integral representation of the Lagrangianfunction and make some useful comments.
PACS numbers: 45.05.+ x ; 02.30.Zz ; 02.03.HqKeywords:
Calculus of variation; Inverse problem; Lagrangians; Linear and nonlinear systems
1. Introduction
In point mechanics the term ‘analytic representation’ refers to description of Newtonian systems by means ofLagrangians [1]. Understandably, to find the analytic representation of a mechanical system one begins with theequation of motion and then constructs a Lagrangian function by using a strict mathematical procedure discovered byHelmholtz [2, 3]. In the calculus of variation this is the so-called inverse variational problem which is more complicatedthan the usual direct problem where one first assigns a Lagrangian function using phenomenological considerationand then computes the equation of motion using the Euler-Lagrange equation [4]. However, there are two typesof analytic representation, namely, the direct and indirect ones. We can introduce the basic concepts of direct andindirect analytic representations by using a system of two uncoupled harmonic oscillators with equations of motion¨ q ( t ) + ω q ( t ) = 0 (1)and ¨ y ( t ) + ω y ( t ) = 0 . (2)It is straightforward to verify that the system of equations (1) and (2) can be analytically represented either by theLagrangian L d = 12 ( ˙ q ( t ) + ˙ y ( t )) − ω q ( t ) + y ( t )) (3)or by the Lagrangian L i = ˙ q ( t ) ˙ y ( t ) − ω q ( t ) y ( t ) . (4)Here overdots denote differentiation with respect to time t . The function L d refers to a Lagangian that gives the directanalytic representation of the system presumably because it yields the equation of motion for q ( t ) ( y ( t )) via the Euler-Lagrange equation written in terms of q ( t ) ( y ( t )). On the other hand, L i yields the equation of motion for q ( t ) ( y ( t ))via the Euler-Lagrange equation written in terms of y ( t ) ( q ( t )). This is why the representation of the system by theuse of L i is called indirect analytic representation. This simple example indicates that Lagrangian representations of ∗ Electronic address: binoyt123@rediffmail.com
Newtonian systems are not unique. The problem of non-uniqueness of the Lagrangian functions has deep consequencesfor the correspondence between symmetries and constants of the motion. For example, the direct Lagrangian (3)is rotationally invariant such that the associated Noether constant of the motion is the angular momentum. Asopposed to this, the indirect Lagrangian (4) is invariant under ’squeeze’ transformation ( q ( t ) , y ( t ) → q ( t ) e t , y ( t ) e − t ).Consequently, for the Lagrangian L i , conservation of angular momentum is associated with the invariance undersqueeze [5].In this work we shall first examine how Hemholtz conditions are useful to study analytic representation of Newtoniansystems modeled by linear second-order ordinary differential equations and then provide a general method to constructtheir direct and indirect analytic representations by using the theory of self-adjoint differential equations [6]. Inparticular, we show that the self-adjoint form of a Newtonian equation can always be used to find its direct analyticrepresentation even if the original equation does not satisfy the Helmholtz criteria [2, 3]. On the other hand, anyNewtonian equation in conjunction with its adjoint permits one to construct its indirect analytical representation.We shall demonstrate the simplicity and effectiveness of our approach by presenting a number of case studies.The general concept of self-adjointness for linear differential equations is well documented in the mathematicalliterature [7, 8]. This is, however, not the case with the nonlinear equations although there have been some attemptsto build a theory of nonlinear self-adjointness [9, 10]. It appears that it will not be straightforward to use this theoryto study the analytic representation of nonlinear systems. However, in the recent past the nonlinear variationalproblem has been studied [11–13] employing an integral representation [14] of the Lagrangian function. The integralrepresentation was derived by applying the Cauchy’s method of characteristics [15] to solve the equation satisfiedby the first integral of an N-dimensional autonomous system. From (1), (2) and (4) it is obvious that the indirectanalytic representation of a linear system is obtained by doubling the number of degrees of freedom. It will be quitesignificant to examine if the same is also true for uncoupled nonlinear equations. Admittedly, a straightforward wayto achieve this consists in using the integral representation of the Lagrangian function sought in ref. 14.In Sec. 2 we indicate how one can find the Lagrangian representation of a Newtonian system when its equationof motion satisfies Helmholtz condition. We adapt, in Sec.3, the theory of self-adjoint second-order linear differentialequations to provide a complete solution for the inverse problem of the calculus of variation. With special attentionto the damped Harmonic oscillator we obtain both direct and indirect analytic representation of the system. Wedevote Sec. 4 to present results for direct and indirect analytic representation for some linear second-order ordinarydifferential equations of mathematical physics. In Sec. 5 we consider a two dimensional autonomous system (linearor nonlinear), provide an integral representation for its Lagrangian function and subsequently, make use of the resultto derive analytic representations of a number of nonlinear Newtonian equations. We find some differences betweenthe analytic representations of uncoupled linear and nonlinear equations. The coupled nonlinear systems, however,exhibit properties which are formally similar to those of the corresponding linear equations. Finally, in Sec. 6 wesummarize our outlook on the present work and make some concluding remarks.
2. Helmholtz conditions and Lagrangian representation
From the inverse problems in the calculus of variations [1] one knows that all Newtonian systems cannot haveLagrangian representation. In particular, the equations written in the general form F i = A ij ( t, q, ˙ q )¨ q j + B i ( t, q, ˙ q ) = 0 , i, j = 1 , , ....., n and q = q ( t ) ∈ R n (5)will have a Lagrangian representation if and only if ∂F i ∂ ¨ q j = ∂F j ∂ ¨ q i , (6a) ∂F i ∂ ˙ q j + ∂F j ∂ ˙ q i = 2 ddt ( ∂F i ∂ ¨ q j ) (6b)and ∂F i ∂q j − ∂F j ∂q i = 12 ddt ( ∂F i ∂ ˙ q j − ∂F j ∂ ˙ q i ) . (6c)The relations (6a)-(6c) are often called Helmholtz conditions [2, 3] and give the necessary and sufficient conditionsfor the existence of a Lagrangian function for any Newtonian system. Equation (5) represents an n dimensionaldifferential equation. For the one-dimensional case (6a) and (6c) become identity such that we are now left with onlyone condition ∂F∂ ˙ q = ddt ( ∂F∂ ¨ q ) . (7)The general equation for the one-dimensional Newtonian system (linear) can be written as F = ¨ q ( t ) + r ( t ) ˙ q ( t ) + s ( t ) q ( t ) = 0 . (8)Equation (8) will satisfy the Helmholtz condition (7) if r ( t ) = 0. Thus we have¨ q + s ( t ) q ( t ) = 0 . (9)Multiplying (9) by δq and integrating over t from t to t we can recast it in the variational form δ Z t t ( 12 ˙ q ( t ) − s ( t )2 q ( t )) dt = 0 (10)such that (9) follows from the Lagrangian function L = 12 ˙ q ( t ) − s ( t )2 q ( t ) (11)via the Euler-Lagrange equation ddt ( ∂L∂ ˙ q ( t ) ) − ∂L∂q ( t ) = 0 . (12)
3. Inverse problem using the theory of self-adjoint differential equations
Here we shall take recourse to the use of the theory for self-adjoint differential equations to construct Lagrangianrepresentation of Newtonian systems. For arbitrary values of r ( t ) and s ( t ), (8) is not self-adjoint. However, the theoryof linear second-order self-adjoint differential equations is quite general [5]. For example, we can always transform (8)in the self-adjoint form F sadj by multiplying it with a non-vanishing factor ρ ( x ) = e R r ( t ) dt (13)such that F sadj = ddt ( ˙ q ( t ) e R r ( t ) dt ) + s ( t ) q ( t ) e R r ( t ) dt = 0 . (14)On the other hand, the adjoint equation F adj corresponding to (8) can be found by changing the dependent variableby q ( t ) = y ( t ) e − R r ( t ) dt . (15)We thus have F adj = ¨ y ( t ) + b ( t ) ˙ y ( t ) + c ( t ) y ( t ) = 0 (16)with b ( t ) = − r ( t ) and c ( t ) = − ˙ r ( t ) + s ( t ) . (17)Understandably, the equation will be self-adjoint if F = F adj .Multiplying (14) by δq ( t ) and integrating over t we can recast it in the variational form δ Z t t L d ( q ( t ) , ˙ q ( t ) , t ) dt = 0 (18)such that L d = e R r ( t ) dt ( ˙ q ( t )2 − s ( t ) q ( t )2 ) . (19)We now make use of (8) and its adjoint (16) to construct a Lagrangian for the system as L i = y ( t ) F + q ( t ) F adj − ddt ( y ( t ) ˙ q ( t )) − ddt ( q ( t ) ˙ y ( t )) . (20)In writing (20) we have assumed that the Lagrangian of a system is its own equation of motion [4, 11]. The third andfourth terms represent the trivial gauge terms [16, 17] for a second-order Lagrangian and have been introduced onlyto write a first-order Lagrangian for the system. Equation (20) on simplification reads L i = ˙ q ( t ) ˙ y ( t ) + r ( t )2 ( q ( t ) ˙ y ( t ) − y ( t ) ˙ q ( t )) + ( 12 ˙ r ( t ) − s ( t )) q ( t ) y ( t ) . (21)It is now straightforward to verify that the explicitly time-dependent Lagrangian L d in (19) gives a direct analyticrepresentation of (8), while the explicitly time-independent Lagrangian L i in (21) gives its indirect analytic represen-tation. This confirms that the theory of self-adjoint differential equations provides us with a unique mathematicalframework to study the Lagrangian structure of any Newtonian system modeled by general linear second-order ordi-nary differential equation which may or may not satisfy the Helmholtz condition [2, 3]. As a very instructive examplefor the effectiveness of the approach derived, we first consider the variational properties for the linearly dampedHarmonic oscillator.The equation of motion for the damped Harmonic oscillator can be written as F = m ¨ q ( t ) + γ ˙ q ( t ) + kq ( t ) = 0 , (22)where m and k stand for the mass and spring constant of the oscillator and the symbol γ represents the frictionalcoefficient of the medium in which the oscillation takes place. The term γ ˙ q ( t ) accounts for the dissipation of energyfrom the system to the environment. The dissipative or non-conservative systems were found not to follow naturallyfrom the Hamilton’s variational principle. Consequently, there have been long standing efforts to construct actionfunctionals for such systems. The first proposition in respect of this was made by Rayleigh [4] who introduceda dissipation function D = γ ˙ x ( t ) in addition to the usual Harmonic oscillator Lagrangian L to express (22) inthe variational form. This proposition did not receive much attention presumably because, rather than one, herewe require two scalar functions, namely L and D , to write the action principle. Other major propositions include(a) construction of a time-dependent Lagrangian [18–20] directly from (22) and (b) construction of an explicitlytime-independent Lagrangian by doubling the degrees of freedom of the system [21]. Here we shall obtain these time-dependent and time-independent analytic representations by using the theory of self-adjoint differential equationsalone.From (8) and (22) r ( t ) = γ/m and s ( t ) = k/m . Using these values in (14) and (19) we write F sadj = e γtm (¨ q ( t ) + γm ˙ q ( t ) + km q ( t )) = 0 (23)and the Lagrangian L d = e γtm m ( 12 m ˙ q ( t ) − kq ( t )) (24)for the direct analytic representation of the damped Harmonic oscillator. Equation (24) represents the result firstreported in ref. 18. We shall now use (22) and the corresponding adjoint equation to obtain a time-independentLagrangian for the damped Harmonic oscillator. Meanwhile, it will be useful to provide a brief review of the work inref. 21 in which Bateman considered the damped oscillator together with an amplified oscillator such that the energydrained out from the first is completely absorbed by the second. Understandably, we have now a dual system whichis closed. The amplified oscillator associated with the damped system (22) was chosen as F ′ = m ¨ y ( t ) − γ ˙ y ( t ) + ky ( t ) = 0 . (25)Mechanistically, this attempt to understand dissipation by the simultaneous use of (22) and (25) amounts to doublingthe degrees of freedom to study the problem. Using F and F ′ we write a Lagrangian given by L i = ( y ( t ) F + q ( t ) F ′ − m ddt ( y ( t ) ˙ q ( t ) + q ( t ) ˙ y ( t ))) / . (26)In the simplified form (26) reads L i = m ˙ q ( t ) ˙ y ( t ) + γ q ( t ) ˙ y ( t ) − ˙ q ( t ) y ( t )) − kq ( t ) y ( t ) . (27)The explicitly time-independent result in (27) provides an indirect analytic representation of the damped harmonicoscillator.From the above it is evident that in order to bring the damped Harmonic oscillator within the framework of varia-tional principle, the image equation (25) was introduced by Bateman [21] using purely phenomenological arguments.It is an interesting mathematical curiosity to note that (25) represents the adjoint equation of (22). This can easily beproved by making use of (16) and (17). Thus we see that, from mathematical point of view, the amplified oscillatorof Bateman is represented by the adjoint of the damped Harmonic oscillsator.From the self-adjoint forms of (22) and (25) it is also possible to find a direct Lagrangian L d = e γtm m ( 12 m ˙ q ( t ) − kq ( t )) + e − γtm m ( 12 m ˙ y ( t ) − ky ( t )) (28)for the Bateman dual system.We conclude by noting that (8) with prescribed analytical properties of the real valued coefficients r ( t ) and s ( t ) oversome region of interest a ≤ t ≤ b represents the most general linear second-order homogeneous ordinary differentialequation. The self-adjoint form of the equation leads rather naturally to its direct analytic representation. On theother hand, the indirect analytic representation of the equation can be obtained by combining it with the associatedadjoint equation.
4. Analytic representation of some special second-order differential equations
Here we make use of formalism of Sec. 3 to obtain results for direct and indirect Lagrangians of a number ofequations which have important applications in physical theories. In Tables I and II we present results for seven suchequations. In particular, Table I gives the results for direct Lagrangian while Table II contains similar results forthe indirect Lagrangian. In presenting the results we always use t as the independent variable and, for brevity, callit as time. Column 2 in Table I gives the self-adjoint equations corresponding to the original equations in column1.The expressions for the Lagrangian are presented in column 3. The Legendre equation is self-adjoint such that Original differential equation Self-adjoint equation Lagrangian giving directanalytic representationLegendre:(1 − t )¨ q ( t ) − t ˙ q ( t ) + n ( n + 1) q ( t ) = 0 (1 − t )¨ q ( t ) − t ˙ q ( t ) + n ( n + 1) q ( t ) = 0 (1 − t ) ˙ q ( t ) − n ( n + 1) q ( t )Bessel: t ¨ q ( t ) + t ˙ q ( t ) + ( t − ν ) q ( t ) = 0 t ¨ q ( t ) + ˙ q ( t ) + ( t − ν t ) q ( t ) = 0 t ˙ q ( t ) − ( t − ν t ) q ( t )Laguerre: t ¨ q ( t ) + (1 − t ) ˙ q ( t ) + nq ( t ) = 0 e − t ( t ¨ q ( t ) + (1 − t ) ˙ q ( t ) + nq ( t )) = 0 te − t ˙ q ( t ) − ne − t q ( t )Hermite:¨ q ( t ) − t ˙ q ( t ) + 2 nq ( t ) = 0 e − t (¨ q ( t ) − t ˙ q ( t ) + 2 nq ( t )) = 0 e − t ˙ q ( t ) − ne − t q ( t )Chebyshev:(1 − t )¨ q ( t ) − t ˙ q ( t ) + n q ( t ) = 0 − √ t − ((1 − t )¨ q ( t ) − t ˙ q ( t ) + n q ( t )) = 0 √ t − q ( t ) + √ t − n q ( t )Gaussian Hypergeometric:(1 − t ) t ¨ q ( t ) + ((1 + α + β ) t − γ ) ˙ q ( t ) (1 − t ) − − α − β + γ t − − γ ((1 − t ) t ¨ q ( t )+ t − γ (1 − t ) − α + β + γ ( ˙ q ( t ) − αβq ( t ) = 0 ((1 + α + β ) t − γ ) ˙ q ( t ) − αβq ( t )) = 0 + αβq ( t )2 t (1 − t ) )Confluent Hypergeometric: t ¨ q ( t ) + ( γ − t ) ˙ q ( t ) − αq ( t ) = 0 e − t t c − ( t ¨ q ( t ) − ( γ − t ) ˙ q ( t ) − αq ( t )) = 0 t γ e − t ( ˙ q ( t ) + t αq ( t ))TABLE I: Self-adjoint differential equations and Lagrangians giving direct analytic representation of some important linearsecond-order differential equation of mathematical physics. the equations in columns 1 and 2 of row 1 are same. On the other hand, the other equations in the table are non-self-adjoint. Consequently, for these equations the self-adjoint forms are different from the parent equations. AllLagrangians giving the direct analytic representation are explicitly time dependent and closely resemble the result in(24) for the damped harmonic oscillator.In close analogy with the results displayed in Table I we reserve columns 1 and 2 of Table II for the originalequations and their adjoints. In column 3 we present results for Lagrangians giving indirect analytic representationfor the same set of equations as considered in Table I. Looking closely into the entries of Table II we see that theLegendre equation and its adjoint are same. This result is quite expected since Legendre equation is self-adjoint. TheLagrangian function for the Legendre equation is of the same form as that in (27) for the damped harmonic oscillatorexcept that the Lagrangian does not involve any term analogous to the middle term in (27). This is, however, nottrue for other equations in the table, which are not self-adjoint. For example, the Lagrangian functions for all otherequations have middle terms in the form β ( t ) = y ( t ) ˙ q ( t ) − q ( t ) ˙ y ( t ). Original differential equation Adjoint equation Lagrangian giving indirectanalytic representationLegendre:(1 − t )¨ q ( t ) − t ˙ q ( t ) + n ( n + 1) q ( t ) = 0 (1 − t )¨ y ( t ) − t ˙ y ( t ) + n ( n + 1) y ( t ) = 0 (1 − t ) ˙ q ( t ) ˙ y ( t ) − n ( n + 1) q ( t ) y ( t )Bessel: t ¨ q ( t ) + t ˙ q ( t ) + ( t − ν ) q ( t ) = 0 t ¨ y ( t ) + 3 t ˙ y ( t ) + (1 + t − ν ) y ( t ) = 0 t ˙ q ( t ) ˙ y ( t ) + t ( y ( t ) ˙ q ( t ) − q ( t ) ˙ y ( t )) − ( t − ν + ) q ( t ) y ( t )Laguerre: t ¨ q ( t ) + (1 − t ) ˙ q ( t ) + nq ( t ) = 0 t ¨ y ( t ) + (1 + t ) ˙ y ( t ) + n ( n + 1) y ( t ) = 0 t ˙ q ( t ) ˙ y ( t ) + t ( y ( t ) ˙ q ( t ) − q ( t ) ˙ y ( t )) − ( n + ) q ( t ) y ( t )Hermite:¨ q ( t ) − t ˙ q ( t ) + 2 nq ( t ) = 0 ¨ y ( t ) + 2 t ˙ y ( t ) + 2( n + 1) y ( t ) = 0 ˙ q ( t ) ˙ y ( t ) + t ( y ( t ) ˙ q ( t ) − q ( t ) ˙ y ( t )) − (2 n + 1) q ( t ) y ( t )Chebyshev:(1 − t )¨ q ( t ) − t ˙ q ( t ) + n q ( t ) = 0 (1 − t )¨ y ( t ) − t ˙ y ( t ) − (1 − t ) ˙ q ( t ) ˙ y ( t )+(1 − n ) y ( t ) = 0 t ( q ( t ) ˙ y ( t ) − y ( t ) ˙ q ( t ))+( − n ) q ( t ) y ( t )Gaussian Hypergeometric:(1 − t ) t ¨ q ( t ) + ((1 + α + β ) t − γ ) ˙ q ( t ) (1 − t ) t ¨ y ( t ) + (2 − t (5 + α + β ) + γ ) ˙ y ( t ) (1 − t ) t ˙ q ( t ) ˙ y ( t )+ − αβq ( t ) = 0 − (3 + α + β + αβ ) y ( t ) = 0 (1 − t − αt − βt + γ ) × ( ˙ q ( t ) y ( t ) − ˙ y ( t ) q ( t ))+ (3 + α + β + αβ ) q ( t ) y ( t )Confluent Hypergeometric: t ¨ q ( t ) + ( γ − t ) ˙ q ( t ) − αq ( t ) = 0 t ¨ y ( t ) + (2 + t − γ ) ˙ y ( t )+ t ˙ q ( t ) ˙ y ( t ) + (1 + t − γ ) × (1 − α ) y ( t ) = 0 ( y ( t ) ˙ q ( t ) − q ( t ) ˙ y ( t )) − ( − α ) q ( t ) y ( t )TABLE II: Adjoint differential equations and Lagrangians giving indirect analytic representation of the linear second-orderdifferential equations in Table I. All examples in Tables I and II represent homogeneous differential equations. It will be instructive to examine ifthe corresponding non-homogeneous equations are also Lagrangian. This is important because the non-homogeneousterm incorporates the effects of the external force on the physical system. Moreover, the equations in the tables alsoplay a crucial role in off-energy-shell potential scattering [22–24]. In the following we make use of a simple systemto demonstrate that the non-homogeneity of differential equations does not bring in any serious complications toconstruct their analytic representation.We are interested in a forced oscillator whose equation of motion is given by¨ q ( t ) + ω q ( t ) = F ( t ) , (29)where F ( t ) stands for an external force acting on the system. It is straightforward to transform (29) in the variationalform so as to obtain the Lagrangian L d = 12 ˙ q ( t ) − ω q ( t ) + q ( t ) F ( t ) . (30)By considering (29) in conjunction with an associated equation¨ y ( t ) + ω y ( t ) = F ( t ) (31)we can write a Lagrangian L i = ˙ q ( t ) ˙ y ( t ) − ω q ( t ) y ( t ) + q ( t ) F ( t ) + y ( t ) F ( t ) (32)for the indirect analytic representation of the system. The simple method presented above is quite general and canalso be used to deal with other non-homogeneous equations which appear in refs. 22, 23, 24 and similar studies.Lagrangians are called standard if they can be expressed as differences between ‘kinetic energy terms’and ‘potentialenergy terms’. The standard Lagrangians, in general, do not depend explicitly on time. In some cases, however, theassociated Lagrangians may depend explicitly on time through exponential factors. The damped harmonic oscillatorprovides a typical example in this respect. The other forms of Lagrangians are referred to as non-standard ones.Linear Newtonian systems require to satisfy Helmholtz conditions to have a standard Lagrangian representation.The violation of Helmholtz conditions, however, does not provide a constraint for having non-standard Lagrangianrepresentations of such systems [25]. The work of Chandrasekar et al. [26] provides a typical example in respect ofthis although their results can be obtained by using a relatively simple mathematical approach [27] and althoughsimilar works appear to have an old root in the classical-mechanics literature [28, 29]. However, it is remarkable thatthe work in ref. 26 was envisaged before the term ‘non-standard Lagrangian’was coined by Musielak [30] and someof the results of Chandrasekar et al. for Hamiltonian functions were re-derived by Bender et al. [31] without takingrecourse to the use of the extended Prelle-Singer method [32, 33].We recognize that studies in Lagrangian and Hamiltonian structures of mechanical systems starting from oneconstant of the motion or first integral of the corresponding equation of motion are of considerable current interest[14, 26, 34]. In this context, an interesting question arises: Can the equations of motion themselves, rather than theirfirst integrals, be used to provide Lagrangian description of mechanical systems? This important issue was consideredby Hojman [35] who derived a very satisfactory method to obtain Lagrangian representation of linear Newtoniansystems using their equations of motion. The basic philosophy of his approach is based on the following symmetryconsideration.The relation between symmetries of a Lagrangian and conserved quantities of the corresponding equation of motionis provided by the so-called Noether’s theorem and is a very well known result in classical mechanics. In contrastto this, it is less well known that the symmetries of the equations of motion form a larger set than the symmetriesof the Lagrangian. However if s-equivalence (a Lagrangian symmetry in which several constants of the motion maybe associated with one symmetry transformation) is taken into account the set of Lagrangian symmetries coincideswith that of the equation of motion [36]. This is perhaps the reason why Hojman [35] could find a short circuit toconstruct Lagrangians from the equations of motion rather than taking recourse to the use of their first integrals.However, instead of going into the details of the work in ref. 35, we shall apply the method to construct a Lagrangianrepresentation for a system of explicitly velocity-dependent two-dimensional differential equations given by¨ x + γ ˙ x + ω x = αy (33)and ¨ y − γ ˙ y + ω y = αx. (34)Here x = x ( t ) and y = y ( t ). Henceforth we shall follow this notation throughout. Physically, (33) and (34) representtwo coupled Harmonic oscillators embedded in a dissipative medium of frictional coefficient γ . Here α stands for thecoupling constant. In the language of Bateman [21] (33) represents a damped oscillator while the oscillator in (34) isan amplifier that absorbs energy drained out from that in (33). For the coupled system (33) and (34), we introducethe Lagrangian L = y (¨ x + γ ˙ x + ω x − αy ) + x (¨ y − γ ˙ y + ω y − αx ) − ddt ( y ˙ x + x ˙ y ) (35)which finally gives L = ˙ x ˙ y + 12 γ ( x ˙ y − y ˙ x ) + 12 α ( x + y ) − ω xy. (36)It is straightforward to see that (36) provides an indirect analytic representation of our coupled system and also thatthe system cannot have a direct analytic representation. More significantly, one can verify that if both equations in(33) and (34) would represent damped oscillators, it would not be possible to construct an analytic representation forthe coupled equations. Curiously enough, the undamped system corresponding to (33) and (34) can have both directand indirect analytic representations [11].So far we have studied the direct and indirect analytic representations of linear Newtonian systems in some detail.We shall now envisage a similar study for nonlinear systems with a view to illustrate the points of contrast and ofsimilarity between the linear and nonlinear problems in respect of their analytic representations. In close analogy withthe works in refs. 11, 12 and 14 we begin the next section with a two dimensional autonomous differential equation(linear or nonlinear) and provide a general method to find its Lagrangian from the constant of the motion. We thenapply this approach to a number of physically important nonlinear equations.
5. Lagrangians for second-order nonlinear differential equations
The general form of a two-dimensional differential equation (linear or nonlinear) can be written as¨ x i = f i ( x j , ˙ x j ) , i, j = 1 , . (37)Equivalently, (37) reads dv i dt = f i ( ~x, ~v ) (38)with ~x = ( x , x ) , ~v = d~xdt = ( v , v ) . (39)A constant of the motion, K ( ~x, ~v ), of (37) or (38) satisfies X i =1 ( f i ( ~x, ~v )) ∂K∂v i + v i ∂K∂x i = 0 (40)along the integral curve of the equation. The solutions or integral surfaces of (40) can be obtained from the equationof characteristics [15] dv f ( ~x, ~v ) = dv f ( ~x, ~v ) = dx v = dx v (41)and subsequently used to write the Lagrangian of (37) as an integral representation [11, 12, 14] L ( ~x, ~v ) = 12 X i =1 v i Z v i K ( i ) n ( ~x, ξ ) ξ dξ. (42)The subscript n in K has been used to differentiate between various constants of the motion that result from (41).Understandably, (42) represents an integral representation of the Lagrangian function.Let us now make use of (41) and (42) to construct an indirect analytic representation for the nonlinear equation d xdt + sin x = 0 . (43)To that end we consider (43) together with an associated equation d ydt + sin y = 0 . (44)Admittedly, this amounts to doubling the number of degrees of the system represented by (43). For (43) and (44) wecan write the equation of characteristics as dv x − sin x = dv y − sin y = dxv x = dyv y . (45)Equation (45) can be arranged in two different ways to get v x dv x + v y dv y = − sin xdx − sin ydy (46)and v x dv y + v y dv x = − sin xdy − sin ydx. (47)We can integrate (46) to obtain the constant of the motion K ( ~x, ~v ) = 12 ( v x + v y ) − cos x − cos y. (48)Following the prescription given in ref. 11 we can construct from (48) the results for K (1)1 ( . ) and K (2)1 ( . ) which occurin (42). We thus obtain L = 12 ( v x + v y ) + cos x + cos y (49)to provide a direct analytic representation for nonlinear oscillator system (43) and (44). One would, therefore, expectthat the constant of the motion found from (47) will lead to an indirect analytic representation. But unfortunately,this is not possible because (47) cannot be integrated to write an expression for the constant of the motion. The simpleexample considered here concludes that the indirect analytic representation of an uncoupled nonlinear system cannotbe found by doubling the number of degrees of freedom. One can verify that this conclusion is true for arbitrarynonlinear equations. In the small angle limit (sin x = x, sin y = y ) our nonlinear system becomes identical to thelinear system as given in (1) and (2). In this case (47) can be integrated to find a constant of the motion K ( ~x, ~v ) = v x v y + xy (50)to obtain an indirect analytic representation as noted in (4).In the above context we note that Chandrasekar et al. [37] while predicting some unusual nonlinear properties ofthe Li´enard-type oscillator ¨ x + kx ˙ x + k x + λ x = 0 (51)made use of the extended Prelle-Singer method to obtain a direct Lagrangian for it. We demonstrate that a relativelysimpler analytic representation of (51) can be found by writing it in the autonomous form v ( x ) ddx v ( x ) + kxv ( x ) + k x + λ x = 0 , v ( x ) = dxdt . (52)The result for the first integral of (51) can be obtained by solving (52) to read K ( x, v ) = h/g (53)with h = (9 λ + k x + 3 kxv ) (54)and g = 9 λ + k x + 9 kxv. (55)From (42) and (53) we get the required Lagrangian as L = 3 k ˙ x log(9 λ + k x + 6 k ˙ x ) − k x . (56)The result in (56) provides a direct analytic representation for Li´enard-tytpe oscillator. Since (51) involves a velocity-dependent or dissipative term, it may be of some interest to follow Bateman [21] and introduce a dual system¨ y − ky ˙ y + 19 k y + λ y = 0 (57)0with a view to look for an indirect analytic representation for the equation in (51). We have verified that, as opposedto the linear dissipative systems, no such representation exists for the Li´enard type oscillator and this is true for otheruncoupled nonlinear dissipative equations.We next consider the coupled Duffing oscillators given by [38]¨ x + ω x + 4 αx + 12 αxy = 0 and ¨ y + ω y + 4 αy + 12 αx y = 0 (58)which represent an important nonlinear system that plays a role in many applicative contexts including the detectionof machinery faults [39]. For the differential equations in (58), the equation of characteristics (41) can be arranged intwo different ways so as to obtain the following results for the constants of the motion given by K = 12 (1 + c ) ˙ x + ω x + y ) + α ( x + y ) + 6 αx y (59)and K = c ˙ x + ω xy + 4 αx y + 4 αxy , (60)where c = ˙ y/ ˙ x that remains constant while constructing the corresponding Lagrangians by the use of (59) and (60)in (42). The equations (59) and (42) give the direct Lagrangian L d = 12 ( ˙ x + ˙ y ) − ω x + y ) − α ( x + y ) − αx y (61)for the coupled system in (58). On the other hand, (60) and (42) lead to L i = ˙ x ˙ y − ω xy − αx y − αxy , (62)the so-called indirect Lagrangian of the system.In close analogy with (33) and (34) for the coupled damped Harmonic oscillators we introduce the damped Duffingoscillators as ¨ x + γ ˙ x + ω x + 4 αx + 12 αxy = 0 and ¨ y − γ ˙ y + ω y + 4 αy + 12 αx y = 0 (63)and verify that the system in (63) can have the indirect analytic representation only given by L i = ˙ x ˙ y − ω xy − γ xy − x ˙ y ) − α ( x + xy ) . (64)The system of equations in (63) has been used to model Soret driven Benard convection [40], vibration of stretchedstring [41] and motion of nonlinear circular plates [42].
6. Concluding remarks
Representation of dynamical systems by Lagrangians or the so-called analytic representation plays a key role indiverse areas of physics ranging from classical mechanics to quantum field theory. In general, for any given system onecan solve the inverse variational problem to construct either the direct or indirect analytic representation. Mechanicalsystems can also admit both representations simultaneously. The most common example in respect of this is providedby the damped harmonic oscillator. In this work we have explicitly demonstrated that self-adjoint form of the equationof motion provides a basis to construct direct analytic representation. On the other hand, the original Newtonianequation and its adjoint taken together can be used to derive the so-called indirect analytic representation. Ourapproach to the inverse problem clearly shows that how, without taking recourse to the use of phenomenologicalarguments, one can efficiently employ the theory of second-order differential equation to bring open systems withinthe framework of the action principle. We have first examined this by dealing with the damped harmonic oscillatorand then presented a number of case studies.In recent years there has been resurgence of interest in the Lagrangian and Hamiltonian description of dissipativesystems [43, 44]. The canonical quantization of damped Harmonic oscillator using the indirect Lagrangian representa-tion have been found to be quite straightforward [45–47] because the corresponding Hamiltonian is time independent.The Hamiltonians corresponding to the results for indirect Lagrangians in Table II are, however, not time independent.Thus it will be interesting to derive a quantization procedure for systems characterized by time-dependent indirectLagrangians.1We noted that it is not straightforward to use the theory of nonlinear self-adjointnes [9, 10] for solving the inversevariational problem of nonlinear differential equations. We thus made use of an integral representation of the La-grangian function [14] to compute results for the direct and indirect Lagrangians of a number of physically importantnonlinear systems. Interestingly, we found that, as opposed to the result obtained for an uncoupled linear equation,the indirect Lagrangian for the corresponding nonlinear equation cannot be constructed by doubling the number ofdegrees of freedom of the system. On the other hand, the coupled equations, whether linear or nonlinear, can haveboth direct and indirect analytic representations. The corresponding dissipative systems, however, follow from indirectLagrangians only. If the time evolution of a mechanical system is governed by linear differential equations, ordinaryor partial, the solution of the problem can be studied confidently because linear analysis is based on the assumptionthat individual effects can be unambiguously traced back to particular causes. This assumption does not hold goodfor nonlinear analysis such that patching simple pieces to understand the whole simply does not work. In fact, ascompared to linear systems, the causal information flow in nonlinear ones is highly complicated [48]. It may, therefore,be tempting to attribute the observed anomaly between the analytic representations of linear and nonlinear equations(uncoupled) to the difference in causal relation in these systems. But it remains an interesting curiosity to providean explanation for why the analytic representations of both linear and nonlinear coupled system exhibit identicalbehavior. Moreover, one may also like to extend our method to construct Lagrangians for isochronous Li´enard-typeoscillators of different dimensions [49, 50]. [1] R M Santilli,
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