Department of Physics, Visva-Bharati University, Santiniketan , India 2

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1 On the non-star Lagrangian equations Aparna Saha Benoy Talukar Department of Physics, Visva-Bharati University, Santiniketan 7335, Inia Nutan Palli, Bolpur 7304, Inia . Abstract. In an attempt to look for the root of nonstar Lagrangians in the theories of the inverse variational problem we introuce a logarithmic Lagrangian (LL) in aition to the so-calle reciprocal Lagrangian (RL) that exists in the literature. The equation of motion that follows from the RL coul easily be solve by using its first or Jacobi integral. This is not, however, true for the similar equation resulting from the LL. We make use of a metho of factorization to fin the particular solutions for equations following from both RL an LL subsequently erive a novel approach to obtain their general solutions from the particular ones. The case stuies presente by us inclue the moifie Emen-type equation for which we also construct a new expression for the Lagrangian function (NL). We point out that NL is not relate to RL by a total erivative or gauge term. In spite of that NL leas to the same equation of motion as that given by the RL. PACS numbers : 45.0., Jj, 0.30.Hq, a Keywors : Nonstar Lagrangians, inverse variational problem, reciprocal logarithmic Lagrangians, corresponing equations of motion, physical relevance, general solutions.introuction For conservative systems, there is an elegant formulation of classical mechanics known as the Lagrangian formulation []. The Lagrangian function, L, for a system is efine to be the ifference between the kinetic potential energies expresse as a function of generalize coorinates, q i (t), velocities, (t). Here the overot enotes ifferentiation with respect to time t. For a system of N particles q q, q,... q } q q, q,... q }. i { N i q i { N The Lagrangian function satisfies the so-calle Euler- Lagrange equation given by L L 0. () t q q i i The equations that result from application of () to a particular Lagrangian are the so-calle equations of motion. This is the irect problem of the calculus of variation which provies a framework for the erivation of (). As oppose to this, one can efine an inverse problem [] that begins with the equation of motion then constructs a Lagrangian function consistent with the variational principle. While the irect problem is the conventional one, we nee to make use of highly sophisticate mathematical techniques [3] to eal with the inverse problem. Fortunately, there exists a solution of the inverse problem which is intermeiate in complexity between the irect conventional inverse problems. This solution is obtaine by coupling the Jacobi integral [] with the metho of characteristics [4] as applie to first-orer partial ifferential equations. Here the Lagrangian is constructe using the first integral of the equation of motion rather than the equation of motion. The first integral is also calle the constant of the motion. The Lagrangian function L ( x, x ) for a one-imensional system having a constant of the motion c( x, x ) can be foun from [5] x c( x, ) L( x, x ) x. () Many physical information of a system are encoe in the Lagrangian functions [6]. Also these functions in conjunction with Ritz optimization proceure [7] can be use to fin approximate analytical solutions of physically important nonlinear evolution equations [8]. These facts have motivate a large number of stuies on the irect inverse problems of Lagrangian mechanics. In the recent past, Cariñena, Rañaa Santer [9] introuce the concept of non-natural Lagrangians which involve neither the kinetic term nor the potential function but lea to equations of motion that represent physically interesting nonlinear systems. All results were presente without explicit reference to the inverse

2 problem of the calculus of variations. Relatively recently, Musielak [0], Cieśliński Nikiciuk [] use a kin of Bootstrap Moel to construct expressions for star (kinetic energy minus potential energy) as well as non-star (nonnatural) Lagrangians (NSLs) with a view to ientify classes of equations of motion that amit a Lagrangian representation. In ref. the non-star Lagrangians of ref.9 were terme as reciprocal Lagrangians. The object of the present work is to look into the root of non-star Lagrangians in the theories of inverse variational problem by making use of () also to provie an uncomplicate metho to solve the equations of motion that follow from them. In section we make use of () to introuce the socalle reciprocal Lagrangian. Interestingly, the propose solution of the inverse problem introuces in a rather natural way a logarithmic type of NSLs. We fin that both reciprocal logarithmic Lagrangians via Euler-Lagrange equations lea to Liénar-type nonlinear ifferential equations which iffer in their constant coefficients. By calculating Jacobi integrals [] we show that equations resulting from the reciprocal Lagrangian are easily solvable. This is, however, not true for equations following from the logarithmic Lagrangian. In section 3 we envisage some case stuies obtain a few equations of physical interest. In this context we also present a new result for the NSL of the moifir Emen-type equations []. In section 4 we provie solutions of the equations by using an uncomplicate metho. Finally, in section 5 we make some concluing remarks with a view to summarize our outlook of the present work.. Non-star Lagrangians For the equation x ( t ) 0 representing the motion of a free particle, both c / x ( t) c x ( t) can be regare as constants of the motion. From the solution of the inverse variational problem in (), the Lagrangians corresponing to c c can be obtaine as L ( x, x ) / x L ( x, x ) x ln x. It may be note that Lagrangians x ln x ln x via Euler-Lagrange equation lea to the same equation of motion. Thus we shall work with L ( x, x ) / x L ( x, x ) ln x (3) As a simple eformation of (3) we introuce the Lagrangians L ( x, x ) /( x f ( x)) (4a) ( x, x ) ln( x f ( )) (4b) L x where f ( x ) is a well-behave function of x. Clearly, (4a) sts for the inverse Lagrangian while (4b) represents the corresponing logarithmic one. Thus we see that the root of the nonstar Lagrangians lies in (3) which was obtaine by using the techniques of inverse variational problem. The form in (4a), first introuce in Ref.9, was applie to the stuy of nonlinear systems relate to the generalize Riccati equations. Therefore, it will be interesting to stuy the properties of the equations of motion that result from the new Lagrangian in (4b). The results in (4a) (4b) via the Euler-Lagrange equation lea to the equations of motion 3 x xf ff 0 (5a) x xf ff 0 (5b) respectively. The prime on f enotes ifferentiation with respect to x. Both equations represent Liénartype nonlinear systems. One may try to solve them by using their first integrals. In orer that we calculate the Jacobi integrals [] corresponing to L (.) L (, ), fin f x J ( f x ) (6a) x J ln( f f x x ). (6b) From these equations we see that (6a) as obtaine using L (.) can be inverte to write x as a function of J x while (6b) involves x essentially in non-algebraic way. It is rather curious to note that although the ifferential equations (5a) (5b) are of the same type their first integrals iffer in analytic properties. 3. Case stuies : NSLs equations of motion We shall consier here the particular forms of the nonstar Lagrangians (4a) (4b), construct some physically important equations of motion from them. For the choice f kx with k, a timeinepenent constant, the reciprocal logarithmic Lagrangians lea to n

3 3 n n x knx x nk x 0, (7a) J n n x knx x nk x 0 (7b) n kx x, (8a) n ( kx x ) x n J ln( kx x ). (8b) n kx x Here n,,3... etc.. For n the equations of motion Jacobi integrals are given by 3 x kx k x 0, (9a) x kx k x 0 (9b) kx x J ( kx x ), (0a) x ln( kx ). (0b) kx x J x Clearly, both (9a) (9b) represent equations of motion for the ampe harmonic oscillator; the first system is over-ampe secon one is critically ampe. These equations can be solve by simple quarature. The solution of (9a) may also be foun from the Jacobi integral. But it is not possible to obtain the solution (8b) using (9b). This serves as a typical example that a solvable equation cannot be solve using its first integral. It is well known that the Lagrangian for the ampe harmonic oscillator is explicitly time epenent [3]. In the past there were many attempts to construct the time-inepenent Lagrangian function for issipative systems thus accommoate them in the frame of action principle in the usual manner. For example, in a remarkable work Bateman [4] foun a time-inepenent Lagrangian by oubling the egree of freeom of the system. Relatively recently, the same problem was stuie by taking recourse to the use of fractional calculus [5]. It is remarkable that the non-star Lagrangians from which the issipative equations (9a) (9b) followe are explicitly time inepenent by choice. For n the equations of motion Jacobi integrals are given by 3 x 3kx x k x 0, (a) 3 x 4kx x k x 0 (b) J kx x, (a) ( kx x ) x J ln( kx x ). (b) kx x Both (0a) (0b) st for the moifie Ementype equations [] of the form x xx x 3 0 (3) with for (a) for (b). 9 8 Equations (a) (b) appear in a number of applicative contexts. For example, they occur in the stuy of equilibrium configurations of spherical gas clou uner the mutual attraction of its molecules subject to the laws of thermoynamics [6] in the moeling of the fusion of pallets [7]. A set of first integrals for (a) (b), other than the Jacobi integrals in (a) (b), can be obtaine by writing each of them as a system of couple first-orer ifferential equations. For example, (a) is equivalent to x y, y y (t ) (4a) 3. (4b) y 3kxy k x.combining (4a) (4b) we fin 3 y y 3kxy k x 0. (5) Here, as before, the prime enotes ifferentiation with respect to x. This equation can be integrate to write the constant of the motion as c( x, x ) x kx. (6) x kx From () (6) we get a new result L NL x kx. (7) The Lagrangian in (7) is not connecte to the reciprocal one by a gauge term but both of them give the same equation of motion. Such Lagrangians are often calle alternative Lagrangians. Two important physical consequences for the existence of alternative Lagrangians of a physical system are (i) there can arise ambiguities in the association of symmetries 3

4 conserve quantities (ii) ifferent Lagrangians can give rise to wiely ifferent Hamiltonian systems such that the same classical system can lea to entirely ifferent quantum mechanical systems [8]. For (b) we foun the constant of the motion as c( x, kx x ) ln( x kx ). (8) x kx From () (8) we verifie that the Lagrangian corresponing to this first integral is the same as that foun from (4b). Thus, as oppose to (a), (b) oes not amit an alternative Lagrangian representation. 4. Solutions of the linear nonlinear equations In an interesting work Péréz Rosu [9] showe that for polynomial nonlinearities particular solutions of Liénar-type ifferential equations can be foun by factorizing the secon-orer system into two firstorer operators. Reyes Rosu [0] foun that it is a nontrivial problem to construct the general solution from the particular one.we shall erive an uncomplicate metho to fin general solutions of our equations by the use of factorization metho. The generic form of the Liénar-type ifferential equation is given by x g ( x ) x F ( x ) 0 (9) where g ( x ) F ( x ) are well efine polynomials in x. In the factorization metho of fining a particular solution of (9) one first writes it in the form ( D ( x))( D ( x)) x 0, D (0) t such that x ( x) x 0. () x Comparison of (9) () yiels g ( x ) ( x ) (a) x F ( x ). (b) x The values of are now chosen with some pre-factors or multiplicative constants so as to satisfy (b). Subsequently, the multiplicative constants in the s are etermine by taking recourse to the use i ' of (a). The particular solution of (9) is then obtaine from the first-orer ifferential equation ( D ) x 0. (3) We now escribe our metho to construct the general solution from a given particular primitive. Before applying it to complicate nonlinear equations we woul like to illustrate the proceure by ealing with the exactly solvable equations in (9a) (9b). For (9a), equations corresponing to (a) (b) rea 3 ( x ) k (4) x k. (5) Let us assume ak a k (6) where a is a constant. From (4) (6) we have a. Using the value a in we solve (3) to get a particular solution of (9a) in the form kt x ce (7) where c is a constant of integration. In the spirit of the metho of variation of parameters we now allow the constant c in (7) to become a function of time t [] such that kt x c( t) e. (8) This value of x when substitute in (9a) leas to the ifferential equation c kc 0 (9) for c (t ). The solution of (9) can now be substitute in (8) to get kt kt x Ae Be (30) 4

5 where A B are the constants of integration arising from the solution of (9). A similar consieration when applie to (9b) gives the solution 4 e ( kt k e ) c ( t ) (37) ( t ) kt x ( A Bt ) e. (3) Note that using a we woul get the same results as those in (30) (3). The first two nonlinear equations in the hierarchy of equations provie by (7a) (7b) are the moifie Emen-type equations in (a) (b). We shall now apply the proceure outline above to obtain their solutions. For (a) we have 3kx (3a) x k x. (3b) In view of (3b) we choose akx kx (33) a with a an arbitrary constant. Equations (3a) (33) can be combine to get a. For a, (3) gives the particular solution x, (34) kt c where c is a time-inepenent constant. We now make c time epenent write where are new constants of integration. If we take 0 we get kt c( t ) (38) t which when use in (34) leas to well-known solution t x. (39) kt The moifie Emen-type equation with possesses eight parameter Lie point 9 symmetries [].This might be the reason why our metho worke so smoothly to obtain the solution. This is, however, not the case with (b) where. Thus a useful check on the fielity of our 8 metho will consist in applying it to (b). It is straightforwar to verify that the particular solution of (b) obtaine by the factorization metho is the same as that foun in (34) for (a).to get the general solution we thus substitute (35) in (b) to write ( c kt ) c c 8kc 8k 0. (40) Unlike (36), (40) cannot be integrate analytically. We thus try to fin a solution of it by introucing a change of variable written as x. (35) kt c( t ) c( t ) g ( t ) kt. (4) From (40) (4) we get Substituting (35) in (a) we obtain a ifferential equation for c (t ) as ( kt c) c c kc 0. (36) Albeit nonlinear, (36) can be solve analytically to obtain gg g 4kg k 0 (4) which is again of the same form as that in (40). However, if we choose to work with g 0, (4) can be solve to get 5

6 c c i e k ( t c ) 4 erf 4 k g ( t ) e (43) x( t ) ierf exp[ ( k ( z ) { erf ( z )} ] (47) where c c are arbitrary constants of integration, erf sts the inverse error function. Since g 0 implies that c k, the reciprocal of the result in (43) refers only to a solution of (b) which is not the most general one. In the recent past Chrasekar et.al [3] solve (0b) by treating it as a Hamiltonian system then taking recourse to the use of a canonical transformation. Interestingly, our solution of (0b) for c c it is in exact agreement with 0 that of (6) in ref. where it was also not taken as a general solution. The general solution was foun by substituting the result in the equation for the canonical transformation of x (t ). We, however, follow a ifferent viewpoint to fin the general solution. From (35) (4) we see that represents g ( t ) the solution of (b) only when the first erivative of the variational parameter is fixe at k. The question is how can one relax this conition fin the most general solution of the equation. It is easy to verify that the transformation x( t) ln y( t) (44) k t reuces (a) in the linear form y 0 (45) the solution of (45) when substitute in (44) gives the exact solution of (a). Unfortunately, no such transformation can be foun to reuce (b) in the linear form. In this situation we suggest that the choice y ( t ) g ( t ) (46) together with (45) will at least lea to an approximate general solution of (b). Following this viewpoint we obtain where ikt z exp( ). (48) k The solution (47) of (b) is in agreement with that foun in ref.3 where the authors claime their result as an exact one. Thus the analysis presente in this section firmly establishes that the factorization technique supplemente by the metho of variation of parameters as propose by us provies a powerful approach to solve important Newtonian equations that follow from both RL LL type nonstar Lagrangians. 5. Concluing remarks All funamental equations of moern physics can be erive from the corresponing Lagrangian functions. This irect problem of the calculus of variations is well ocumente in many textbooks monographs [4]. On the other h. many physicists are harly familiar with the inverse problem of Lagrangian mechanics []. As we have alreay note, here one begins with the equations of motion then constructs their Lagrangians by a strict mathematical proceure [5]. The general solution of the inverse vaiational was given by Helmholtz [6]. Darboux [7] prove that in the case of one-imensional problems the Lagrangian always exists. But the Lagrangian escription is highly nonunique provies an opportunity to choose Lagrangian functions in a hoc fashions. The nonstar Lagrangian of the reciprocal type [9] provies a typical example in respect of this. In this work we envisage a stuy to fin the root of the RL in the theory of inverse variational problem. This provie us an opportunity to introuce another non-star Lagrangian, namely, the logarithmic Lagrangian which we abbreviate as LL. In writing the result for the LL we have chosen to work with the eformation of c ln x rather than c x ln x. Because of the non-uniqueness of the Lagrangians for one-imensional systems, this particular choice is not a limitation of our approach. Also we note that Lagrangians obtaine from the eformation of c x ln x lea to physically uninteresting 6

7 equations of motion. However, we foun that the complete escription of physical systems require the knowlege of both RL LL. We emonstrate this with particular emphasis on the equations of the ampe harmonic oscillator Moifie Ementype equation. The latter represents a nonlinear orinary ifferential equation for which it is ifficult to evise a star metho of solution. We have emonstrate that these equations can be satisfactorily solve by taking recourse to the use of a metho of factorization in conjunction with a variant of the Lagrange s metho of variation of parameters. A very interesting feature of the factorization metho is that the set of equations foun from the use of RL LL has exactly the same particular solution. But their general solutions are quite ifferent. We conclue by noting that Lagrangian formalism evelope by us is base a efinitive solution of the inverse variational problem of classical ynamics. Our metho of solving the nonlinear equations which result the erive Lagrangians is also quite straightforwar for application to other equations of nonlinear ynamics. References [] Golstein H 998 Classical Mechanics 7th reprint (Narosa Pub. House, New Delhi, Inia) [] Santilli R M 978 Founations of theoretical mechanics I (Springer-Verlag, New York) First Eition; Hojman S Urrutia L F 98 On the inverse problem of the calculus of variations J. Math. Phys [3] Olver P J 993 Applications of Lie Groups to Differential Equations (Springer-Verlag, New York) [4] Sneon I N 957 Elements of partial ifferential equations (McGraw Hill, New York) [5] Lopez G 996 Hamiltonians Lagrangians for N-imensional autonomous systems Ann. Phys. (NY) [6] Hojman S 984 Symmetries of Lagrangians of their equations of motion J. Phys. A: Math. Gen ; Ibragimov N H Kolsru T 003 Lagrangian approach to evolution equations : Symmetries conservation laws Nonlinear Dynamics [7] Arfken G B Weber H J 004 ( Elsevier Pvt. Lt. New Delhi, Inia) [8] Golam Ali Sk Talukar B 009 Couple BEC solitons in optical lattices Ann. Phys. (NY) 85-99; Pal Debabrata, Golam Ali Sk Talukar B 0 Stability of embee solitons in higher-orer NLS equations Physica Scr (5); Golam Ali Sk, Maria Salerno, Talukar B, Saha Aparna 0 Displace Dynamics of binary mixtures in linear nonlinear optical lattices Phys. Rev. A () [9] Cariñena J F, Rañaa M F Santer F 005 Lagrangian formalism fot nonlinear secon-orer Riccati equations : One-imensional integrability two-imensional super interability J. Math. Phys (7) [0] Musielak Z E 008 Star non-star Lagrangians for issipative ynamical systems with variable coefficients J. Phys. A: Math. Theor (7) [] Cieśliński J I Nikiciuk T 00 A irect approach to the construction of star nonstar Lagrangians for issipative-like ynamical systems with variable coefficients J. Phys. A: Math. Theor (4) [] Ince E L 956 Orinary Differential Equations (Dover, New York) [3] Calirola P 94 Froze non-conservative nella meccanica quantistica Nuovo. Cim ; Kanai E 948 On the quantization of issipative systems Prog. Theor. Phys [4] Bateman H 93On the issipative systems relate variational principles Phys. Rev [5] Riewe F 996 Non-conservative Lagrangian Hamiltonian mechanics Phys. Rev. E [6] Chrasekhar S 957 An Introuction to the Stuy of Stellar Structure (Dover Publication, New York) [7] Erwin V J, Ames W F Aams E 984 Wave Phenomena : Moern Theory Applications e C Rogers J B Mooie ( North Holl Pub. Co. : Amsteram) [8] Mori G, Ferrario C, Vecchio G L, Marmo G Rubano C 990 The inverse problem of the calculus of variations the geometry of the tangent bunle Phys. Rep [9] Cornejo Péréz O Rosu H C 005 Nonlinear secon- orer oe s : Factorization particular solutions Prog. Theor. Phys [0] Reyes M A Rosu H C 008 Riccati parameter solutions of nonlinear ODEs J. Phys. A: Math. Theor [] Babistar A W 967 Transcenental function satisfying non-homogeneous linear ifferential equations (McMillan, New York) [] Leach P G 985 First integrals of the moifir Emen equation n q ( t) q q 0 J. Math. Phys [3] Chrasekar V K, Senthilvelan M Lakshmanan M 007 On the general solution for the moifie Emen-type equation x xx x 3 0 J. Phys. A [4] Morse P M Feshbach 953 Methos of Theoretical Physics (McGraw Hill, New York); 7

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