Noether versus Killing symmetry of conformally flat Friedmann metric

In a recent study Noether symmetries of some static spacetime metrics in comparison with Killing vectors of corresponding spacetimes were studied. It was shown that Noether symmetries provide additional conservation laws that are not given by Killing vectors. In an attempt to understand how Noether symmetries compare with conformal Killing vectors, we find the Noether symmetries of the flat Friedmann cosmological model. We show that the conformally transformed flat Friedman model admits additional conservation laws not given by the Killing or conformal Killing vectors. Inter alia, these additional conserved quantities provide a mechanism to twice reduce the geodesic equations via the associated Noether symmetries.     

Symmetries of Type D Pure Radiation Fields

The geometrical symmetries corresponding to the continuous groups of collineations and motions generated by a null vector l are considered. These symmetries have been translated into the language of Newman-Penrose formalism for pure radiation (PR) type D fields. It is seen that for such fields, conformal, special conformal and homothetic motions degenerate to motion. The concept of free curvature, matter curvature and matter affine collineations have been introduced and the conditions under which PR type D fields admit such collineations have been obtained. Moreover, it is shown that the projective collineation degenerate to matter affine, special projective, conformal, special conformal, null geodesic and special null geodesic collineations. It is also seen that type D pure radiation fields admit Maxwell collineation along the propagation vector l.     

Projective differential geometry and geodesic conservation laws in general relativity,II: Conservation laws

We examine the geodesic conservation laws associated with the projective actions discussed in our earlier paper with the same overall title. Using the Cartan formalism, a one-to-one correspondence between a class of these actions and all geodesic conservation laws is possible. In particular there is a natural geometric interpretation of Killing tensors. Homothetic motions are shown to correspond to conserved quantities on all geodesies (not just null ones). The same approach identifies homothetic Killing tensors and a universal quadratic first integral which reduces to the conformai Killing tensor case on null geodesics.     

Conservation Laws and Associated Noether Type Vector Fields via Partial Lagrangians and Noether’s Theorem for the Liang Equation

We show how one can construct conservation laws of the Liang equation which is not variational but may be regarded as Euler-Lagrange in part. This first requires the determination of the Noether-type symmetries associated with the partial Lagrangian. The final construction of the conservation laws resort to a formula equivalent to Noether’s theorem. A variety of subclasses are given and, for each, a large number of conserved flows are found—the method is usable for any general choice of the variable speed of sound.     

Variational principle for theP(4) affine theory of gravitation and electromagnetism

We propose a Lagrangian for theP(4) theory of gravitation and electromagnetism which is a straightforward generalization of the Einstein Lagrangian. A constrained Palatini variation of this Lagrangian yields the geometrical Einstein-Maxwell affine field equations. We show that these results can be extended easily to include both electric and magnetic charges. Finally, we consider conservation laws arising from the invariance properties of the Lagrangian.     

Projective symmetries and laws of conservation in the K-spaces determined by gravitational fields

The symmetries of equations of motion for probe bodies (projective symmetries) and the corresponding laws of conservation in the K-spaces determined by the gravitational fields of type (3) are studied. The results define all mechanical and field laws of conservation in the foregoing gravitational fields resulting from projective symmetries, in particular, from isometries and homotheties. The metric ansatzes found can be used for construction of new exact solutions to the Einstein equations and for examination of their large-scale (geodesic) structure. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 30–37, April, 2008.     

Symmetries,integrals, and three-dimensional reductions of Plebañski’s second heavenly equation

We study symmetries and conservation laws for Plebañski’s second heavenly equation written as a first-order nonlinear evolutionary system which admits a multi-Hamiltonian structure. We construct an optimal system of one-dimensional subalgebras and all inequivalent three-dimensional symmetry reductions of the original four-dimensional system. We consider these two-component evolutionary systems in three dimensions as natural candidates for integrable systems.     

Comment on “Classification of Cosmic Scale Factor via Noether Gauge Symmetries” [Int. J. Theor. Phys. 54, 2343 (2015)]

We discuss the relationship between the Noether point symmetries of the geodesic Lagrangian, in a (pseudo)Riemannian manifold, with the elements of the Homothetic algebra of the space. We observe that the classification problem of the Noether symmetries for the geodesic Lagrangian is equivalent with the classification of the Homothetic algebra of the space, which in the case of a Friedmann-Lemaître-Robertson-Walker spacetime is a well-known result in the literature.     

Noether-Symmetry Analysis using Alternative Lagrangian Representations

We have sought to work with an approach to Noether symmetry analysis which uses the properties of infinitesimal point transformations in the space-time (q, t) variable to establish the association between symmetries and conservation laws of a dynamical system. In this approach symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of two uncoupled Harmonic oscillators and two such oscillators coupled by an interaction. Both these systems can have alternative Lagrangian representations. We have studied in detail how the association between symmetries and conservation laws changes as one alters the analytic or Lagrangian representation. This analysis is carried out with a view to explicitly demonstrate that the correlation between symmetry transformation and corresponding invariant quantity depends crucially on the choice of the analytic representation. PACS 45.20.Jj, 45.20.df, 45.20.dh     

Variational Formulation of Approximate Symmetries and Conservation Laws

We show how the conserved vectors and associated (approximate) Lie symmetry generators of a partial differential equation with a small parameter can be utilized to construct approximate Lagrangians for the equation. We then use the Lagrangian to further determine approximate Noether symmetries and, hence, new associated conservation laws. The theory is applied to a number of perturbations of the wave equation.     

Gauge invariance,charge conservation,and variational principles

We present new results on the correspondence between symmetries, conservation laws and variational principles for field equations in general non-abelian gauge theories. Our main result states that second order field equations possessing translational and gauge symmetries and the corresponding conservation laws are always derivable from a variational principle. We also show by the way of examples that the above result fails in general for third order field equations.     

Energy conditions and conservation laws in LTB metric via Noether symmetries

In this paper, we have investigated Noether symmetries in Lemaitre–Tolman–Bondi (LTB) metric. Using the Lagrangian associated with the LTB metric, the set of determining equations for Noether symmetries is obtained and then integrated in several cases. It is shown that the LTB metric can be classified in to eight distinct classes corresponding to Noether algebra of dimension 4, 5, 6, 7, 8, 9, 11 and 17. The obtained Noether symmetries are compared with Killing and homothetic vectors. The well known Noether’s theorem is used to find the expressions for conservation laws in each case. Moreover, it is shown that most of the obtained metrics are anisotropic or perfect fluid models which satisfy certain energy conditions and the equation of state.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

Potential Symmetries and Associated Conservation Laws to Fokker-Planck and Burgers Equation

Kara and Mahomed have derived an identity, which does not rely on use of a Lagrangian as needed to obtain conservation laws by Noethers theorem. By using the identity and symbolic computation, conservation laws arising from nonlocal symmetries are obtained for Fokker-Planck equation and burgers equation.     

Geometric aspects of some hidden symmetries

Hidden symmetries of two dimensional chiral models are analysed from the geometric point of view. The dual symmetry gives rise to generalized isometries of the metric on the space of dependent variables. The Jacobi equation of geodesic deviation is dual invariant and the generalized isometries lead to generalized symmetries of the field equations. Being variational divergence symmetries they generate families of conservation laws.     

Symmetries,Conservation Laws and Multipliers via Partial Lagrangians and Noether’s Theorem for Classically Non-Variational Problems

We show how one can construct conservation laws of equations that are not variational but are Euler–Lagrange in part using Noether-type symmetries associated with partial Lagrangians. These Noether-type symmetries are, usually, not symmetries of the system. The resultant construction of the conservation law resorts to a formula equivalent to Noether’s theorem. A variety of examples are given.     

Relationship between Symmetries andConservation Laws

The fundamental relation between Lie-Bäcklund symmetry generators andconservation laws of an arbitrary differential equation is derived without regardto a Lagrangian formulation of the differential equation. This relation is used inthe construction of conservation laws for partial differential equations irrespectiveof the knowledge or existence of a Lagrangian. The relation enables one toassociate symmetries to a given conservation law of a differential equation.Applications of these results are illustrated for a range of examples.     

Noether theorem in higher-order Lagrangian mechanics

We study higher-order Lagrangian mechanics on thek-velocity manifold. The variational problem gives rise to new concepts, such as main invariants, Zermelo conditions, higher-order energies, and new conservation laws. A theorem of Noether type is proved for higher-order Lagrangians. The invariants to the infinitesimal symmetries are explicitly written. All this construction is a natural extension of classical Lagrangian mechanics.