New non-standard Lagrangians for the Liénard-type equations

Kudryashov, Nikolay A.; Sinelshchikov, Dmitry I.

doi:10.1016/j.aml.2016.07.028

Cited by 16 publications

(12 citation statements)

References 21 publications

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“…A Jacobi multiplier can also be used to solve the inverse problem for a second-order differential equation, allowing us to find a (may be non-standard) Lagrangian description for a given secondorder differential equation when a Jacobi multiplier is known [5,28,31,36], or also constants of motion when two inequivalent Jacobi multipliers are known, leading to the result, usually attributed to Currie and Saletan [18], that if two regular Lagrangians L 1 and L 2 are known for a second-order differential equation, the quotient function f defined by…”

Section: Examplesmentioning

confidence: 99%

“…The above condition (4.4) is known in the literature as Chiellini condition [13], and appears in this example as a consequence of the Jacobi multiplier theory. More details can be found in [4,31].…”

Section: Liénard's Equation and Chiellini's Conditionmentioning

confidence: 99%

“…The Chiellini condition (4.4) is used to integrate the first kind Abel equations [24], and has been applied for obtaining exact solutions of second-order differential equations that can be reduced to an Abel equation. It was shown in [30] the relationship of this condition to its linearisability under an appropriate Sundman transformation [39], and then it was shown in [31] that the corresponding Lagrangian can be easily derived from that of the damped harmonic oscillator.…”

Section: Liénard's Equation and Chiellini's Conditionmentioning

confidence: 99%

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Jacobi multipliers and Hamel’s formalism

Cariñena¹,

Santos²

2021

J. Phys. A: Math. Theor.

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Section: Examplesmentioning

confidence: 99%

Section: Liénard's Equation and Chiellini's Conditionmentioning

confidence: 99%

Section: Liénard's Equation and Chiellini's Conditionmentioning

confidence: 99%

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Jacobi multipliers and Hamel’s formalism

Cariñena¹,

Santos²

2021

J. Phys. A: Math. Theor.

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“…Linearization and equivalence to some Painlevé-Gambier equations via the generalized Sundman transformations were considered in [6, 7, 10-12, 17, 20]. In works [1,13,22,23,29] Lagrangians and Jacobi last multipliers for the Liénard-type equations were studied. As far as integrability of (1) is concerned, its linearizability conditions via both point and Sundman transformations can be obtained from the results of [14,15,32].…”

Section: Introductionmentioning

confidence: 99%

Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

Sinelshchikov

Gaiur

Kudryashov

2019

Journal of Mathematical Analysis and Applications

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We consider a family of non-autonomous second-order differential equations, which generalizes the Liénard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold-Liouville integrability of Hamiltonian systems. We illustrate our results by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation.

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“…Some previously known criteria for integrability were also reobtained, and several new examples of integrable Liénard equations were given. A new family of the Liénard-type equations which admits a non-standard autonomous Lagrangian was found in [18], and autonomous first integrals for each member were obtained. Four new integrability criteria of a particular type of Liénard type equations were obtained in [19] by studying the connections between this family of Liénard-type equations and type III Painlevé-Gambier equations.…”

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confidence: 99%

On the Integrability of the Abel and of the Extended Liénard Equations

Mak

Harkó

2019

Acta Math. Appl. Sin. Engl. Ser.

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We present some exact integrability cases of the extended Liénard equation y ′′ + f (y) (y ′ ) n + k (y) (y ′ ) m + g (y) y ′ + h (y) = 0, with n > 0 and m > 0 arbitrary constants, while f (y), k(y), g(y), and h(y) are arbitrary functions. The solutions are obtained by transforming the equation Liénard equation to an equivalent first kind first order Abel type equation given by dvAs a first step in our study we obtain three integrability cases of the extended quadratic-cubic Liénard equation, corresponding to n = 2 and m = 3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Liénard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Liénard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with g(y) ≡ 0 and h(y) ≡ 0, and arbitrary n and m, thus allowing to obtain the general solution of the corresponding Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.

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New non-standard Lagrangians for the Liénard-type equations

Cited by 16 publications

References 21 publications

Jacobi multipliers and Hamel’s formalism

Jacobi multipliers and Hamel’s formalism

Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

On the Integrability of the Abel and of the Extended Liénard Equations

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