Approximate Noether Symmetries Research Articles

Approximate Noether theorem and its inverse for nonlinear dynamical systems with approximate nonstandard Lagrangian

The approximate Noether theorem and its inverse theorem for the nonlinear dynamical systems with approximate exponential Lagrangian and approximate power-law Lagrangian are investigated. For each case, the approximate differential equations of motion for the nonlinear dynamical systems with approximate nonstandard Lagrangian are established, the generalized Noether identities are given. The relationship between the approximate Noether symmetries and approximate conserved quantities for the system with approximate nonstandard Lagrangian are established, and the approximate Noether theorems and their inverse theorems are obtained. Two examples are given to illustrate the application of the results.

The approximate Noether symmetries and conservation laws for approximate Birkhoffian systems

The approximate Noether symmetries and conservation laws for approximate Birkhoffian systems

Approximate Noether symmetries of the geodetic Lagrangian of spherically symmetric spacetimes

Approximate Noether symmetries of the geodetic Lagrangian of spherically symmetric spacetimes

2nd order approximate Noether and Lie symmetries of Gibbons–Maeda–Garfinkle–Horowitz–Strominger charged black hole in the Einstein frame

In this paper, approximate Noether and Lie symmetries of 2nd order for Gibbons–Maeda–Garfinkle–Horowitz–Strominger (GMGHS) charged black hole in the Einstein frame are analyzed comprehensively. To explore the approximate Noether symmetries of 2nd order, Noether symmetries of Minkowski spacetime are used which forms a 17 dimensional Lie algebra. It is observed that no new approximate Noether symmetry is obtained at 1st and 2nd order. To examine the 1st and 2nd order approximate Lie symmetries of the GMGHS black hole spacetime, 35 Lie symmetries (exact) of the Minkowski spacetime are used which forms an algebra sl(6, R). It is shown that no new approximate Lie symmetry exists at 1st and 2nd order and only exact 35 symmetries are recouped as trivial approximate Lie symmetries at both orders. Furthermore, no energy rescaling factor is seen in this spacetime.

Integration of the geodesic equations via Noether symmetries

Through this paper, I will overview the use of Noether symmetry approach in discussing the integration of geodesic equations for the geodesic Lagrangians of spacetimes. I will also give some examples to reveal the efficiency of Noether symmetry approach by finding the first integrals related for the geodesic Lagrangians of the Gödel-type, Schwarzschild, Reïssner–Nordström and Kerr spacetimes. After obtaining the approximate Noether symmetries of the Schwarzschild, Reïssner–Nordström and Kerr spacetimes, the first integrals associated with each of approximate Noether symmetries have been integrated to find a general solution of geodesic equations in terms of the arc length [Formula: see text].

Approximate Mei Symmetries and Invariants of the Hamiltonian

It is known that corresponding to each Noether symmetry there is a conserved quantity. Another class of symmetries that corresponds to conserved quantities is the class of Mei symmetries. However, the two sets of symmetries may give different conserved quantities. In this paper, a procedure of finding approximate Mei symmetries and invariants of the perturbed/approximate Hamiltonian is presented that can be used in different fields of study where approximate Hamiltonians are under consideration. The results are presented in the form of theorems along with their proofs. A simple example of mechanics is considered to elaborate the method of finding these symmetries and the related Mei invariants. At the end, a comparison of approximate Mei symmetries and approximate Noether symmetries is also given. The comparison shows that there is only one common symmetry in both sets of symmetries. Hence, rest of the symmetries in the two sets correspond to two different sets of conserved quantities.

Approximate Noether Symmetries of Perturbed Lagrangians and Approximate Conservation Laws

In this paper, within the framework of the consistent approach recently introduced for approximate Lie symmetries of differential equations, we consider approximate Noether symmetries of variational problems involving small terms. Then, we state an approximate Noether theorem leading to the construction of approximate conservation laws. Some illustrative applications are presented.

The dynamics of particles around time conformal AdS-Schwarzschild black hole

Abstract In this article, we explore the effect of the cosmological constant ( Λ ) on the dynamics of neutral as well as charged particles in vicinity of the time conformal Schwarzschild black hole. The general time conformal term e ϵ f ( t ) is introduced in the AdS-Schwarzschild spacetime, where f(t) is an arbitrary function of time t and ϵ is a small parameter which causes perturbation in the spacetime. The corresponding Lagrangian is used in the Noether symmetry equation which gives a system of 19 partial differential equations (PDEs). The solution of the 19 PDEs give us the require time conformal AdS-Schwarzschild spacetime along with approximate Noether symmetries and conservation laws. The dynamics of neutral and charged particles are discuss over the fabric of time conformal Ads–Schwarzschild spacetime.

The approximate Noether symmetries and approximate first integrals for the approximate Hamiltonian systems

We provide the Hamiltonian version of the approximate Noether theorem developed for the perturbed ordinary differential equations (ODEs) (Govinder et al. in Phys Lett 240(3):127–131, 1998) for the approximate Hamiltonian systems. We follow the procedure adopted by Dorodnitsyn and Kozlov (J Eng Math 66(1–3):253–270, 2010) for the Hamiltonian systems of unperturbed ODEs. The approximate Legendre transformation connects the approximate Hamiltonian and approximate Lagrangian. The approximate Noether symmetries determining equation for the approximate Hamiltonian systems is defined explicitly. We provide a formula to establish an approximate first integral associated with an approximate Noether symmetry of the approximate Hamiltonian systems. We analyzed several physical models to elaborate the approach developed here.

The Second-Order Correction to the Energy and Momentum in Plane Symmetric Gravitational Waves Like Spacetimes

This research provides second-order approximate Noether symmetries of geodetic Lagrangian of time-conformal plane symmetric spacetime. A time-conformal factor is of the form e ϵ f ( t ) which perturbs the plane symmetric static spacetime, where ϵ is small a positive parameter that produces perturbation in the spacetime. By considering the perturbation up to second-order in ϵ in plane symmetric spacetime, we find the second order approximate Noether symmetries for the corresponding Lagrangian. Using Noether theorem, the corresponding second order approximate conservation laws are investigated for plane symmetric gravitational waves like spacetimes. This technique tells about the energy content of the gravitational waves.

Nth-Order Approximate Lagrangians Induced by Perturbative Geometries

A family of perturbative Lagrangians that describe approximate and multidimensional Klein-Gordon equations are studied. We probe the existence of approximate Noether symmetries via generalized geometric conditions for a perturbation of any order. The knowledge of the geometric conditions uncovers that unlike exact symmetries, the approximate Noether symmetries of the Lagrangian which describes the motion of a particle in n-dimensional space under the action of an autonomous force, is inequivalent to the Noether symmetries admitted by the Klein-Gordon Lagrangian in general.

Correction to the conservation laws in plane symmetric gravitational wave-like spacetimes

This paper discusses the approximate Noether symmetries of the action of plane symmetric spacetime. A time-dependent conformal factor is introduced in the plane symmetric metric in such a way so as to obtain the same number of Noether symmetries as for the exact plane symmetric spacetime. The corresponding approximate conservation is obtained to discuss the conservation laws of the energy and momentum during the formation of gravitational wave.

Approximate Noether symmetries and collineations for regular perturbative Lagrangians

Regular perturbative Lagrangians that admit approximate Noether symmetries and approximate conservation laws are studied. Specifically, we investigate the connection between approximate Noether symmetries and collineations of the underlying manifold. In particular we determine the generic Noether symmetry conditions for the approximate point symmetries and we find that for a class of perturbed Lagrangians, Noether symmetries are related to the elements of the Homothetic algebra of the metric which is defined by the unperturbed Lagrangian. Moreover, we discuss how exact symmetries become approximate symmetries. Finally, some applications are presented.

Cylindrically symmetric gravitational-wavelike space–times

We present Noether symmetries of a geodetic Lagrangian for a time-conformal cylindrically symmetric space–time. We introduce a time-conformal factor in the general cylindrically symmetric space–time to make it nonstatic and then find approximate Noether symmetries of the action of the corresponding Lagrangian. Taking the perturbation up to the first order, we find all Lagrangians for cylindrically symmetric space–times for which approximate Noether symmetries exist.

Particle dynamics around time conformal regular black holes via Noether symmetries

The time conformal regular black hole (RBH) solutions which are admitting the time conformal factor [Formula: see text], where [Formula: see text] is an arbitrary function of time and [Formula: see text] is the perturbation parameter are being considered. The approximate Noether symmetries technique is being used for finding the function [Formula: see text] which leads to [Formula: see text]. The dynamics of particles around RBHs are also being discussed through symmetry generators which provide approximate energy as well as angular momentum of the particles. In addition, we analyze the motion of neutral and charged particles around two well known RBHs such as charged RBH using Fermi–Dirac distribution and Kehagias–Sftesos asymptotically flat RBH. We obtain the innermost stable circular orbit and corresponding approximate energy and angular momentum. The behavior of effective potential, effective force and escape velocity of the particles in the presence/absence of magnetic field for different values of angular momentum near horizons are also being analyzed. The stable and unstable regions of particle near horizons due to the effect of angular momentum and magnetic field are also explained.

Approximate Noether symmetries from geodetic Lagrangian for plane symmetric spacetimes

Noether symmetries from geodetic Lagrangian for time-conformal plane symmetric spacetime are presented. Here, time-conformal factor is used to find the approximate Noether symmetries. This is a generalization of the idea discussed,5–6 where they obtained approximate Noether symmetries from Lagrangian for a particular plane symmetric static spacetime. In the present paper, the most general plane symmetric static spacetime is considered and perturbed it by introducing a general time-conformal factor eϵf(t), where ϵ is very small which causes the perturbation in the spacetime. Taking the perturbation up to the first-order, we find all Lagrangian for plane symmetric spacetimes for which approximate Noether symmetries exist.

Energy Content of Colliding Plane Waves Using Approximate Noether Symmetries

We study the energy content of colliding plane waves using approximate Noether symmetries. For this purpose, we use the approximate Lie symmetry method for Lagrangians for differential equations. We formulate the first-order perturbed Lagrangian for colliding plane electromagnetic and gravitational waves. In both cases, we show that no nontrivial first-order approximate symmetry generator exists.

Approximate conservation laws of perturbed partial differential equations

This paper presents a general result on approximate conservation laws of perturbed partial differential equations. A method of constructing approximate conservation laws to systems of perturbed partial differential equations is given, which is based on approximate Noether symmetries of approximate and standard adjoint systems of the original system. The relationship between the Noether symmetry operators of approximate and standard adjoint system is established. As a result, the approach is applied to the perturbed wave equation and the perturbed KdV equation.

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

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 non- trivial 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–Nordstrom metric.

Approximate First Integrals of a Chaotic Hamiltonian System

Approximate first integrals (conserved quantities) of a Hamiltonian dynamical system with two-degrees of freedom which arises in the modeling of galaxy have been obtained based on the approximate Noether symmetries for the resonance ω 1 = ω 2. Furthermore, Kolmogorov-Arnold-Moser (KAM) curves have been obtained analytically and they have been compared with the numerical ones on the Poincaré surface of section.