Noether’s theorem and Ermakov systems for nonlinear equations of motion

Ray, John R.; Reid, James L.

doi:10.1007/bf02902329

Cited by 25 publications

(15 citation statements)

References 8 publications

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“…−N 2 (t)dt 2 + a 2 (t)dΣ 2 K (IV 45). which however differs from our previous form by no more than a temporal diffeomorphism or clock regraduation.The function N (t) is called the lapse function.…”

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confidence: 60%

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Cosmological aspects of the Eisenhart–Duval lift

et al. 2018

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A cosmological extension of the Eisenhart-Duval metric is constructed by incorporating a cosmic scale factor and the energy-momentum tensor into the scheme. The dynamics of the spacetime is governed the Ermakov-Milne-Pinney equation. Killing isometries include spatial translations and rotations, Newton-Hooke boosts and translation in the null direction. Geodesic motion in Ermakov-Milne-Pinney cosmoi is analyzed. The derivation of the Ermakov-Lewis invariant, the Friedmann equations and the Dmitriev-Zel'dovich equations within the Eisenhart-Duval framework is presented.

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confidence: 60%

“…Finding the symmetries of time-dependent harmonic oscillators generated an extensive literature in the early 1980's, including [43][44][45][46]. A discussion in terms of canonical transformations is in [48].…”

Section: Symmetries As Conformal Killing Isometriesmentioning

confidence: 99%

Cosmological aspects of the Eisenhart–Duval lift

et al. 2018

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“…Ray and Reid studied the generalized Ermakov system in different perspectives through a series of papers. [4][5][6][7][8][9][10][11] Goedert 12 constructed the second constant of the motion for a class of Ermakov-Lewis-Ray-Reid systems, and thus, no extra integration was required to construct the general solution of the systems.…”

Section: Introductionmentioning

confidence: 99%

“…The Ermakov system admits a canonical Hamiltonian [17][18][19] provided it satisfies certain conditions. The coupled Ermakov system 20 and uncoupled Ermakov systems 7,8,21 were analyzed with the aid of the Noether theorem. In other studies, 7,8,21 the Lagrangian description was one dimensional, and the Ermakov-Lewis invariant was not constructed from the associated dynamical Noether symmetry as in the case of Haas and Goedert 18 and Moyo and Leach.…”

Section: Introductionmentioning

confidence: 99%

“…The coupled Ermakov system 20 and uncoupled Ermakov systems 7,8,21 were analyzed with the aid of the Noether theorem. In other studies, 7,8,21 the Lagrangian description was one dimensional, and the Ermakov-Lewis invariant was not constructed from the associated dynamical Noether symmetry as in the case of Haas and Goedert 18 and Moyo and Leach. 20 To explore the history of Ermakov system in more detail, the interested reader is referred to the commentary by Leach and Andriopoulos 22 and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Hamiltonian symmetry classification, integrals, and exact solutions of a generalized Ermakov system

Naz

Mahomed

2020

Math Methods in App Sciences

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We investigate the Hamiltonian symmetry classification, first integrals, and exact solutions of the three‐dimensional generalized Ermakov system expressed in Spherical polar form. We utilize the Hamiltonian version of Noether's theorem to construct Hamiltonian symmetries and first integrals. The three‐dimensional generalized Ermakov system involves an arbitrary function F(θ, ϕ). First, the Hamiltonian symmetry classification is performed for the three‐dimensional generalized Ermakov system. For arbitrary form of the function F(θ, ϕ), four first integrals are established, and only three of these are functionally independent. The Hamiltonian integral theorem yields seven different functional forms of F(θ, ϕ) for which the generalized Ermakov system has additional Hamiltonian symmetries and related first integrals. These functional forms are new and not provided before in the literature. Finally, the exact solutions of the generalized Ermakov system, for arbitrary F(θ, ϕ) as well as for different forms of F(θ, ϕ) are provided with the aid of these derived first integrals.

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Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

Struckmeier¹,

Riedel²

2001

Annalen der Physik

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An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation.The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.

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Noether’s theorem and Ermakov systems for nonlinear equations of motion

Cited by 25 publications

References 8 publications

Cosmological aspects of the Eisenhart–Duval lift

Cosmological aspects of the Eisenhart–Duval lift

Hamiltonian symmetry classification, integrals, and exact solutions of a generalized Ermakov system

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

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