Lie symmetries and Noether symmetries

Leach, P.G.L.

doi:10.2298/aadm120625015l

Cited by 13 publications

(18 citation statements)

References 17 publications

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“…The Noether symmetries are then computed from (14) and finally (16) provides the first integrals corresponding to each Noether symmetry. The reader is guided to [46] for further discussions about this technique and its relations with the so-called Noether symmetries.…”

Section: Noether's Approachmentioning

confidence: 99%

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Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations

Naz

Freire

Naeem

2014

Abstract and Applied Analysis

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Different approaches to construct first integrals for ordinary differential equations and systems of ordinary differential equations are studied here. These approaches can be grouped into three categories: direct methods, Lagrangian or partial Lagrangian formulations, and characteristic (multipliers) approaches. The direct method and symmetry conditions on the first integrals correspond to first category. The Lagrangian and partial Lagrangian include three approaches: Noether’s theorem, the partial Noether approach, and the Noether approach for the equation and its adjoint as a system. The characteristic method, the multiplier approaches, and the direct construction formula approach require the integrating factors or characteristics or multipliers. The Hamiltonian version of Noether’s theorem is presented to derive first integrals. We apply these different approaches to derive the first integrals of the harmonic oscillator equation. We also study first integrals for some physical models. The first integrals for nonlinear jerk equation and the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities are derived. Moreover, solutions via first integrals are also constructed.

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Section: Noether's Approachmentioning

confidence: 99%

“…Equations ( 73) finally result in (46) and, hence, following the same procedure as we did for the direct method, five first integrals (50) are obtained.…”

Section: Noether Approach For a System And Itsmentioning

confidence: 99%

Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations

Naz

Freire

Naeem

2014

Abstract and Applied Analysis

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“…For example, Arnold [1] only proves an easier special case, that is not powerful enough to yield the conservation of energy for autonomous systems. Leach [16,17] deplores that, all too often, weak versions of Noether's theorem are presented as the real thing. There have been some authors that tried to restore Noether's theorem to its rightful power in introductory classical mechanics, as for example Lévy-Leblond [18] and Desloge and Karch [8].…”

Section: Introductionmentioning

confidence: 99%

Revisiting Noether's Theorem on constants of motion

Gorni¹,

Zampieri²

2021

JNMP

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In this paper we revisit Noether’s theorem on the constants of motion for Lagrangian mechanical systems in the ODE case, with some new perspectives on both the theoretical and the applied side. We make full use of invariance up to a divergence, or, as we call it here, Bessel-Hagen (BH) invariance. By recognizing that the Bessel-Hagen (BH) function need not be a total time derivative, we can easily deduce nonlocal constants of motion. We prove that we can always trivialize either the time change or the BH-function, so that, in partic- ular, BH-invariance turns out not to be more general than Noether’s original invariance. We also propose a version of time change that simplifies some key formulas. Applications include Lane-Emden equation, dissi- pative systems, homogeneous potentials and superintegrable systems. Most notably, we give two derivations of the Laplace-Runge-Lenz vector for Kepler’s problem that require space and time change only, without BH invariance, one with and one without use of the Lagrange equation

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“…To understand the integrability of these nonlinear ODEs, through Lie symmetry analysis, attempts have been made to extend Lie's theory of continuous group of point transformations in several directions. A few notable extensions which have been developed for this purpose are (i) contact symmetries [23,21,22,24], (ii) hidden and nonlocal symmetries [36,37,31,25,26,27,28,29,30,33,34,35], (iii) λ-symmetries [20,38,39,40], (iv) adjoint symmetries [10,9,41] and (v) telescopic vector fields [42,40].…”

Section: Introductionmentioning

confidence: 99%

Symmetries of nonlinear ordinary differential equations: The modified Emden equation as a case study

2015

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Abstract. Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries, contact symmetries, hidden symmetries, nonlocal symmetries, λ-symmetries, adjoint symmetries and telescopic vector fields of a second-order ordinary differential equation. We also illustrate the algorithm involved in each method by considering a nonlinear oscillator equation as an example. The connections between (i) symmetries and integrating factors and (ii) symmetries and integrals are also discussed and illustrated through the same example. The interconnections between some of the above symmetries, that is (i) Lie point symmetries and λ-symmetries and (ii) exponential nonlocal symmetries and λ-symmetries are also discussed. The order reduction procedure is invoked to derive the general solution of the second-order equation.

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Lie symmetries and Noether symmetries

Abstract: We demonstrate that so-called nonnoetherian symmetries with which a known first integral is associated of a differential equation derived from a Lagrangian are in fact noetherian. The source of the misunderstanding lies in the nonuniqueness of the Lagrangian.

Cited by 13 publications

References 17 publications

Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations

Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations

Revisiting Noether's Theorem on constants of motion

Symmetries of nonlinear ordinary differential equations: The modified Emden equation as a case study

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