Noetherian first integrals

Flessas, G P

doi:10.2991/jnmp.2008.15.1.2

Cited by 4 publications

(5 citation statements)

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“…(with dummy dependence on the variables Q, Q), and it leads back exactly to the integral constant of motion (16). This trivial ψ of formula (26) however only works for that particular motion t → q(t).…”

Section: The Total Derivative Conditionmentioning

confidence: 93%

“…the function (11) has zero derivative at ε = 0. In either case the constant function t → F (t) + ∂ ε G(ε, t)| ε=0 turns out to be (16).…”

Section: Integral Constants Of Motionmentioning

confidence: 99%

“…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%

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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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Section: The Total Derivative Conditionmentioning

confidence: 93%

“…the function (11) has zero derivative at ε = 0. In either case the constant function t → F (t) + ∂ ε G(ε, t)| ε=0 turns out to be (16).…”

Section: Integral Constants Of Motionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Revisiting Noether's Theorem on constants of motion

Gorni¹,

Zampieri²

2021

JNMP

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“…A notable example of this is to be found in the s-and g-symmetries proposed by Hojman and his collaborators [4,5,6,7,8]. Later it was demonstrated [15] that the first integrals for the examples considered in their papers could easily be obtained by a proper application of Noether's Theorem. A more recent illustration is to be found in the paper by Luo and Jia [19] in their determination of an integral for a problem in relativistic mechanics which, again, could not be obtained by an application of Noether's Theorem.…”

Section: Introductionmentioning

confidence: 93%

Lie symmetries and Noether symmetries

Leach

2012

APPL ANAL DISCRETE M

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“…However, equation ( 42) is nothing else than the Noether condition in Hamiltonian formalism for generalized symmetries (see [27]). In the case of point and contact transformations, this is a well known result [28,29]. Therefore the determination of Hojman conservation quantities in cosmological models, which arise from a Lagrange function, it is equivalent with the application of Noether's Theorem.…”

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confidence: 94%

On the Hojman conservation quantities in Cosmology

Paliathanasis

Capozziello

2016

Physics Letters B

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We discuss the application of the Hojman's Symmetry Approach for the determination of conservation laws in Cosmology, which has been recently applied by various authors in different cosmological models. We show that Hojman's method for regular Hamiltonian systems, where the Hamiltonian function is one of the involved equations of the system, is equivalent to the application of Noether's Theorem for generalized transformations. That means that for minimally-coupled scalar field cosmology or other modified theories which are conformally related with scalar-field cosmology, like f (R) gravity, the application of Hojman's method provide us with the same results with that of Noether's theorem. Moreover we study the special Ansatz. φ (t) = φ (a (t)), which has been introduced for a minimally-coupled scalar field, and we study the Lie and Noether point symmetries for the reduced equation. We show that under this Ansatz, the unknown function of the model cannot be constrained by the requirement of the existence of a conservation law and that the Hojman conservation quantity which arises for the reduced equation is nothing more than the functional form of Noetherian conservation laws for the free particle. On the other hand, for f (T ) teleparallel gravity, it is not the existence of Hojman's conservation laws which provide us with the special function form of f (T ) functions, but the requirement that the reduced second-order differential equation admits a Jacobi Last multiplier, while the new conservation law is nothing else that the Hamiltonian function of the reduced equation.PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x The source for the late-time cosmic acceleration [1][2][3][4] has been attributed to an unidentified type of matterenergy with a negative parameter in the equation of state, the dark energy. The cosmological constant, Λ, is the simplest candidate for dark energy with a parameter in the equation of state, w Λ = −1. However, the cosmological constant suffers from two major problems. They are the fine tuning and the coincidence problems [5,6]. Consequently other dark energy candidates have been introduced, such as cosmologies with time-varying Λ (t), quintessence, Chaplygin gas, matter creation, f (R) gravity and many others. This status of art imposes a discrimination among the various cosmological models and the search for new approaches to find out exact solutions to be matched with data. In particular, one of the issue is that cosmological models should come out from some first principles in order to be related to some fundamental * Electronic address: anpaliat@phys.uoa.gr † Electronic address: leach.peter@ucy.ac.cy ‡ Electronic address: capozziello@na.infn.it theory.Below we restrict our consideration to a spatially flat Friedmann-Robertson-Walker (FRW) spacetime with line element ds 2 = −dt 2 + a 2 (t) dx 2 + dy 2 + dz 2 .(1)Let us start our considerations from cosmological solutions derived from General Relativity (GR). Standard GR provides us with a set of second-order differential equations. Cons...

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Noetherian first integrals

Cited by 4 publications

References 22 publications

Revisiting Noether's Theorem on constants of motion

Revisiting Noether's Theorem on constants of motion

Lie symmetries and Noether symmetries

On the Hojman conservation quantities in Cosmology

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