The projective geometric theory of systems of second-order differential equations: straightening and symmetry theorems

Aminova, A. V.; Aminov, Nail' A-M

doi:10.1070/sm2010v201n05abeh004085

Cited by 24 publications

(40 citation statements)

References 14 publications

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“…a specific dynamical system, is done by means of certain simple differential conditions, which involve the elements of the projective algebra and the function f (x i ,ẋ j ). Similar considerations can be found in [4][5][6][7][8].…”

Section: Introductionsupporting

confidence: 53%

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The geometric nature of Lie and Noether symmetries

2011

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It is shown that the Lie and the Noether symmetries of the equations of motion of a dynamical system whose equations of motion in a Riemannian space are of the formẍ i + i jkẋis an arbitrary function of its argument, are generated from the Lie algebra of special projective collineations and the homothetic algebra of the space respectively. Therefore the computation of Lie and Noether symmetries of a given dynamical system in these cases is reduced to the problem of computation of the special projective algebra of the space. It is noted that the Lie and Noether symmetry vectors are common to all dynamical systems moving in the same background space. The selection of the vectors which are Lie/Noether symmetries for a given dynamical system is done by means of a set of differential conditions involving the vectors and the potential function defining the dynamical system. The general results are applied to a number of different applications concerning (a) The motion in Euclidean space under the action of a general central potential (b) The motion in a space of constant curvature (c) The determination of the Lie and the Noether symmetries of class A Bianchi type hypersurface orthogonal spacetimes filled with a scalar field minimally coupled to gravity (d) The analytic computation of the Bianchi I metric when the scalar field has an exponential potential.

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Section: Introductionsupporting

confidence: 53%

“…where i jk are the connection coefficients in an affine space and P i j 1 ... j m (t, x i ) are smooth polynomials completely symmetric in the lower indices [4,5,[9][10][11]. Equation (3) is quite general and covers most of the standard cases autonomous and non autonomous.…”

Section: The Lie Point Symmetry Conditions In An Affine Spacementioning

confidence: 99%

The geometric nature of Lie and Noether symmetries

2011

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“…quadratic or linear, Hermitian or complex combinations of basic operatorsp and x, etc.) and then solving for the coefficients to fulfill (13). The second step in the method is realizing that finding eigenfunctions φ s (x, t) ofÎ L ,…”

Section: The Lewis-riesenfeld Methods In Light Of the Quantum Arnold Tmentioning

confidence: 99%

“…The idea of using invariants to solve equations is rather old, going back to S. Lie (1883) [11] who showed that a second order differential equation has the maximal group of symmetries if the differential equation is up to third order in the derivative, and the coefficients satisfy certain relations [12,13].…”

Section: Introductionmentioning

confidence: 99%

On the Lewis–Riesenfeld (Dodonov–Man’ko) invariant method

Guerrero

López-Ruiz

2015

Phys. Scr.

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Abstract.We revise the Lewis-Riesenfeld invariant method for solving the quantum timedependent harmonic oscillator in light of the Quantum Arnold Transformation previously introduced and its recent generalization to the Quantum Arnold-ErmakovPinney Transformation. We prove that both methods are equivalent and show the advantages of the Quantum Arnold-Ermakov-Pinney transformation over the LewisRiesenfeld invariant method. We show that, in the quantum time-dependent and damped harmonic oscillator, the invariant proposed by Dodonov & Man'ko is more suitable and provide some examples to illustrate it, focusing on the damped case.

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“…Equation (4.6) is a nonautonomous equation. However, it can be written as the following autonomous system [1] d 2T ds 2 −J 2T σ dJ ds…”

Section: Curl Forces In the Presence Of A Dissipative Forcementioning

confidence: 99%