Canonical forms for systems of two second-order ordinary differential equations

Soh, C. Wafo; Mahomed, F. M.

doi:10.1088/0305-4470/34/13/316

Cited by 22 publications

(36 citation statements)

References 9 publications

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“…The following definitions are known (see [15,18,20,26]). The definition on partial Lagrangian was given in [13].…”

Section: Preliminariesmentioning

confidence: 99%

“…Practical criteria for linearization via invertible transformations for quadratic semi-linear systems of second-order ODEs were recently studied by Mahomed and Qadir [19]. The canonical forms for systems of two second-order ODEs admitting a low-dimensional point symmetry algebra were attempted too Wafo Soh and Mahomed [20]. The equivalence problem for systems of second-order ODEs has been considered by Fels [21].…”

Section: Introductionmentioning

confidence: 99%

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Noether, partial Noether operators and first integrals for a linear system

Naeem

Mahomed

2008

Journal of Mathematical Analysis and Applications

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We obtain Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a system of two linear second-order ordinary differential equations (ODEs) with variable coefficients. The canonical form for a system of two second-order ordinary differential equations is invoked and a special case of this system is studied for both Noether and partial Noether operators. Then the first integrals with respect to Noether and partial Noether operators are obtained for the linear system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. This can give rise to further studies on systems from a partial Lagrangian viewpoint as systems in general do not admit Lagrangians.

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“…The following definitions are known (see [15,18,20,26]). The definition on partial Lagrangian was given in [13].…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Noether, partial Noether operators and first integrals for a linear system

Naeem

Mahomed

2008

Journal of Mathematical Analysis and Applications

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“…it is quite restrictive and in the case of systems of two second-order ODEs there are more than 70 canonical forms [13]. This in itself indicates the difficulty in obtaining and using canonical forms.…”

Section: Copyright C 2004 By C Wafo Soh and F M Mahomedmentioning

confidence: 99%

“…but the system itself is not integrable by quadratures [13]. When the hypotheses of Corollary 1 are satisfied, the approach given here is preferable since it does not require the use of canonical variables.…”

Section: Corollarymentioning

confidence: 99%

Reduction of Order for Systems of Ordinary Differential Equations

Soh¹,

Mahomed²

2004

JNMP

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“…In particular, the symmetry properties have been investigated by several researchers (see, e.g. Prince and Eliezer [1], Sen [2], Damianou and Sophocleous [3], Gorringe and Leach [4], Wafo and Mahomed [5,6] and Naeem and Mahomed [7]). Some paradigms of classical mechanical systems are the free particle and the oscillator systems as well as the Kepler problem and its variants.…”

Section: Introductionmentioning

confidence: 99%

Algebraic linearization criteria for systems of ordinary differential equations

2011

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Algebraic linearization criteria by means of general point transformations for systems of two second-order nonlinear ordinary differential equations (ODEs) are revisited. In previous work due to Wafo Soh and Mahomed (Int. J. Non-Linear Mech. 36:671, 2001) two four-dimensional Lie algebras that result in linearizability in terms of arbitrary point transformation for such systems were studied. Here we consider three more algebras of dimension four that result in linearization. Therefore our results supplement those of Wafo Soh and Mahomed (Int. J. Non-Linear Mech. 36:671, 2001). Moreover, it is shown that these are the only other possibilities for dimension four. Hence we provide the complete algebraic linearization criteria for dimension four algebras. Necessary and sufficient conditions for linearization via invertible maps of a nonlinear to a linear system are given. These are shown to be built up from the Lie algebraic criteria for linearization of scalar second-order ODEs. These results together with very recent work (Bagderina in J. Phys. A, Math. Theor. 43:465201, 2010) give a complete picture on linearizability properties via general point transformations for systems of two second-order ODEs. Furthermore, we provide natural extensions of these algebraic criteria for linearizing arbitrary systems of nonlinear second-order ODEs by means of point transformations. We also obtain algebraic criteria for the reduction of a linear system to the simplest system. Examples from Newtonian mechanics and geodesic equations are presented to illustrate our results.

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Canonical forms for systems of two second-order ordinary differential equations

Cited by 22 publications

References 9 publications

Noether, partial Noether operators and first integrals for a linear system

Noether, partial Noether operators and first integrals for a linear system

Reduction of Order for Systems of Ordinary Differential Equations

Algebraic linearization criteria for systems of ordinary differential equations

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