On the complete integrability of a nonlinear oscillator from group theoretical perspective

Bhuvaneswari, A.; Chandrasekar, V. K.; Senthilvelan, M.; Lakshmanan, M.

doi:10.1063/1.4731238

Cited by 32 publications

(42 citation statements)

References 18 publications

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“…We observe that whilst the Mathews-Lakshmanan nonlinear oscillators [18][19][20][21][22][23][24][29][30][31] are reproduced in case (i) and (ii), some "shifted" Mathews-Lakshmanan nonlinear oscillators are obtained in case (iii) for the same PDM m (x) = 1/ 1 ± λx 2 ; λ ≥ 0. Moreover, to show that the usage of the current methodical proposal is not only limited to oscillator linearization, we discuss (in section 4) the mapping of an isotonic nonlinear oscillator into a PDM deformed isotonic oscillator.…”

Section: Introductionmentioning

confidence: 86%

“…For more details on this issue the reader may refer to [17,22]. Obviously, moreover, equation (2) is a quadratic Liénard-type differential equation (quadratic in terms ofẋ 2 in (2)) which serves as a very interesting model in both physics and mathematics (cf., e.g., the sample of references [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] and related references cited therein).…”

Section: Introductionmentioning

confidence: 99%

“…), but also by the mathematical challenge indulged in the von Roos Hamiltonian. Such a position-dependent mass setting have invigorated a relatively recent and rapid research attention on the PDM concept for classical mechanical systems (cf., e.g., [12][13][14][15][16][17][18][19][20][21][22][23][24][25]) which was readily introduced by Mathews and Lakshmanan [18] back in 1974. Unlike the ordering ambiguity that arises in the kinetic energy term of the PDM von Roos quantum mechanical Hamiltonian, no such ordering ambiguities arise in the classical mechanical PDM Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

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Position-dependent mass Lagrangians: nonlocal transformations, Euler–Lagrange invariance and exact solvability

Mustafa¹

2015

J. Phys. A: Math. Theor.

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A general nonlocal point transformation for position-dependent mass Lagrangians and their mapping into a "constant unit-mass" Lagrangians in the generalized coordinates is introduced. The conditions on the invariance of the related Euler-Lagrange equations are reported. The harmonic oscillator linearization of the PDM Euler-Lagrange equations is discussed through some illustrative examples including harmonic oscillators, shifted harmonic oscillators, a quadratic nonlinear oscillator, and a Morse-type oscillator. The Mathews-Lakshmanan nonlinear oscillators are reproduced and some "shifted" Mathews-Lakshmanan nonlinear oscillators are reported. The mapping of an isotonic nonlinear oscillator into a PDM deformed isotonic oscillator is also discussed.

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Position-dependent mass Lagrangians: nonlocal transformations, Euler–Lagrange invariance and exact solvability

Mustafa¹

2015

J. Phys. A: Math. Theor.

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“…Some of the equations which we have identified as linearizable/integrable through Lie symmetry analysis in this class have also been investigated from other perspectives [2][3][4][5].…”

Section: Introductionmentioning

confidence: 97%

“…In this paper, we intend to classify/identify linearizable and integrable nonlinear ODEs of a general mixed quadratic-linear (inẋ) Liénard type equation [6][7][8][9][10][11][12][13] A(ẍ,ẋ, x) ≡ẍ + f (x)ẋ 2 + g(x)ẋ + h(x) = 0, (2) where f (x), g(x) and h(x) are arbitrary functions of x, which is much more challenging than the study of (1). One can observe that (1) is a subcase of (2).…”

Section: Introductionmentioning

confidence: 99%

Lie point symmetries classification of the mixed Liénard-type equation

et al. 2015

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In this paper we develop a systematic and self-consistent procedure based on a set of compatibility conditions for identifying all maximal (eight parameter) and non-maximal (one and two parameter) symmetry groups associated with the mixed quadratic-and h(x) are arbitrary functions of x. With the help of this procedure we show that a symmetry function b(t) is zero for nonmaximal cases, whereas it is not so for the maximal case. On the basis of this result the symmetry analysis gets divided into two cases, (i) the maximal symmetry group (b = 0) and (ii) non-maximal symmetry groups (b = 0). We then identify the most general form of the mixed quadratic linear Liénard-type equation in each of these cases. In the case of eight-parameter symmetry group, the identified general equation becomes linearizable. In the case of non-maximal symmetry groups the identified equations are all integrable. The integrability of all the equations is proved either by providing the general solution or by constructing timeindependent Hamiltonians.

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A relationship between λ‐symmetries and first integrals for ordinary differential equations

Zhang

2019

Math Methods in App Sciences

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In this paper, we provide some geometric properties of -symmetries of ordinary differential equations using vector fields and differential forms. According to the corresponding geometric representation of -symmetries, we conclude that first integrals can also be derived if the equations do not possess enough symmetries.We also investigate the properties of -symmetries in the sense of the deformed Lie derivative and differential operator. We show that -symmetries have the exact analogous properties as standard symmetries if we take into consideration the deformed cases. KEYWORDSdeformed property, first integrals, Frobenius integrable, symmetries, -symmetries MSC CLASSIFICATION 34A05; 34C14 INTRODUCTIONClassical symmetry groups play an important role in analyzing differential equations. They have been widely used to reduce the order of ordinary differential equations and to reduce the number of independent variables of partial differential equations. However, many of the differential equations we study, especially nonlinear differential equations, do not in general possess enough symmetries. It is therefore essential to devise some kinds of proper extensions of classical Lie method in order to handle more situations. In the last few years, various extensions have been done, and some new classes of symmetries have been introduced. Among these extensions, -symmetries have been introduced by Muriel and Romero 1 based on a new method of prolonging vector fields known as -prolongation, which strictly include symmetries.Although -symmetries are not symmetries in the proper sense, as they do not map solutions into solutions, they can be used to perform reduction procedure via exactly the same method used for standard symmetries. Several usual methods of reduction for ODEs that do not come from symmetries are derived from the existence of -symmetries; one can see Muriel and Romero 1 and relevant references for further details. They can also be viewed as a class of twisted symmetries, and the recently developments can be reviewed in Gaeta. 2 Especially, several relations among -symmetries and first integrals for ordinary differential equations are studied in previous studies 3-11 and references therein. For applications of -symmetries to derive first integrals or conservation laws of dynamical systems or Hamiltonian systems, one can see previous studies 12-14 for example. By a geometrical characterization of -prolongation of vector fields, -symmetries have been extended to partial differential equations, leading to -symmetries that can also be used to derive conservation laws of the equations. 15,16 The investigation of -symmetries has practical advantages. First of all, the determining equations for symmetries may not have nontrivial solutions but the corresponding equations for -symmetries may have. In addition, -symmetries can be algorithmically obtained by using the Jacobi last multipliers. 17 Secondly, there exists an unknown function in the determining equations for -symmetries, so comparing with classical Lie method,...

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On the complete integrability of a nonlinear oscillator from group theoretical perspective

Cited by 32 publications

References 18 publications

Position-dependent mass Lagrangians: nonlocal transformations, Euler–Lagrange invariance and exact solvability

Position-dependent mass Lagrangians: nonlocal transformations, Euler–Lagrange invariance and exact solvability

Lie point symmetries classification of the mixed Liénard-type equation

A relationship between λ‐symmetries and first integrals for ordinary differential equations

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