Geometric Linearization of Ordinary Differential Equations

Qadir, Asghar

doi:10.3842/sigma.2007.103

Cited by 18 publications

(25 citation statements)

References 17 publications

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“…As before, the generator W 0 gives translation in s and always exists for a Lagrangian of the type (16) [30], W 1 = [W 0 , Z 4 ] which is a scaling symmetry in s, t, r that can be used to get rid of the s dependence in the generators given by (31) and (32). This is reasonable as symmetries of a Lagrangian always form a sub-algebra of the symmetries of the Euler-Lagrange (geodesic) equations [31] and the algebra of the Euler-Lagrange equations for Minkowski spacetime is sl(6, R) which is 35 dimensional [32].…”

Section: Symmetries and Approximate Symmetries Of A Lagrangian For Thmentioning

confidence: 97%

Approximate Noether symmetries of the geodesic equations for the charged-Kerr spacetime and rescaling of energy

2009

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Using approximate symmetry methods for differential equations we have investigated the exact and approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime. Taking Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional Lie algebra of conformal isometries of Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter we consider first-order approximate symmetries of the Kerr metric as a first perturbation of the Schwarzschild metric. We then consider the second-order approximate symmetries of the Kerr metric as a second perturbation of the Minkowski metric. The approximate symmetries are recovered for these spacetimes and there are no nontrivial approximate symmetries. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr metric has to be rescaled and the rescaling factor is r -dependent. This re-scaling factor is compared with that for the Reissner-Nordström metric.

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Section: Symmetries and Approximate Symmetries Of A Lagrangian For Thmentioning

confidence: 97%

Approximate Noether symmetries of the geodesic equations for the charged-Kerr spacetime and rescaling of energy

2009

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“…We conclude that in this case we have Finally it follows that the conjecture made in [6], claiming that the Lie symmetries of the geodesics of the maximally symmetric spaces of non-vanishing curvature are only the KVs, is true. Another verification of this conjecture is given in [14].…”

Section: The Case Of a Space Of Constant Non-vanishing Curvaturementioning

confidence: 97%

Lie symmetries of geodesic equations and projective collineations

Tsamparlis

Paliathanasis

2010

Nonlinear Dyn

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“…The Lagrangian that minimizes the arc length in (7) admits only the four KVs given by (8) and (9) along with the generator ∂/∂s, which always exists for a Lagrangian for the geodesic equations [10]. This Lagrangian does not admit the HV and CKV given by (10) and (11).…”

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confidence: 90%

A note on Noether symmetries and conformal Killing vectors

Hussain

2010

Gen Relativ Gravit

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A conjecture was stated in Hussain et al. (Gen Relativ Grav 41:2399, 2009, that the conformal Killing vectors form a subalgebra of the symmetries of the Lagrangian that minimizes arc length, for any spacetime. Here, a counter example is constructed to demonstrate that the above statement is not true in general for spacetimes of non-zero curvature.There is a connection between isometries or Killing vectors (KVs) and symmetries of the geodesic equations (Euler-Lagrange equations) of the underlying spaces [2]. Further KVs always form a subalgebra of conformal Killing vectors (CKVs), homothetic vectors (HVs) and curvature collineations (CCs) etc of spacetimes (for detail see [3]). It is also known that the set of Noether symmetries always contained in the set of symmetries of the corresponding Euler-Lagrange equations [4]. This suggests that to know how the Noether symmetries are related with the spacetime symmetries, e.g. CKVs, HVs, etc.The Minkowski spacetime, which is flat and hence conformally flat, admits 15 CKVs [5]. A Lagrangian for the geodesic equations for this spacetime admits 17 Noether symmetries [1]. These 15 CKVs form a proper subalgebra of the 17 Noether

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Geometric Linearization of Ordinary Differential Equations

Cited by 18 publications

References 17 publications

Approximate Noether symmetries of the geodesic equations for the charged-Kerr spacetime and rescaling of energy

Approximate Noether symmetries of the geodesic equations for the charged-Kerr spacetime and rescaling of energy

Lie symmetries of geodesic equations and projective collineations

A note on Noether symmetries and conformal Killing vectors

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