Two dimensional dynamical systems which admit Lie and Noether symmetries

We prove two theorems which relate the Lie point symmetries and the Noether symmetries of a dynamical system moving in a Riemannian space with the special projective group and the homothetic group of the space respectively. The theorems are applied to classify the two dimensional Newtonian dynamical systems, which admit a Lie point/Noether symmetry. Two cases are considered, the non-conservative and the conservative forces. The use of the results is demonstrated for the Kepler - Ermakov system, which in general is non-conservative and for potentials similar to the H\`enon Heiles potential. Finally it is shown that in a FRW background with no matter present, the only scalar cosmological model which is integrable is the one for which 3-space is flat and the potential function of the scalar field is exponential. It is important to note that in all applications the generators of the symmetry vectors are found by reading the appropriate entry in the relevant tables.Comment: 25 pages, 17 table

Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.Comment: 19 page

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. © 2008 Springer Science+Business Media, Inc

The negative evaluative component within the semantic structure of behaviour verbs in Russian, English and Tatar languages

The article deals with the lexical-semantic group of verbs which describe human behavior using the material of three languages (Russian, English and Tatar) with different structure. The situation "human behavior" implies the existence of a behavior subject, a committed action and a behavior object in respect of which an action takes place. The situation also has an observer who evaluates this action. An observer's evaluation is the main characteristic of the studied group of verbs which distinguishes them from other lexicalsemantic groups. Evaluation is considered as the opinion about a subject, expressing its characteristics in terms of value category. At that the estimation within the meaning of a behavior verb is mainly negative. A negative estimation can be expressed explicitly and implicitly. The deviation from the normal existence undergoes a negative qualification and is perceived as a wrong, a reprehensible behavior. Such human qualities as dishonesty, insincerity, irresponsibility, unnatural behavior, etc. are condemned in these verbs. The violation of standards and norms of behavior causes the most severe emotional, verbal and cogitative reactions among the representatives of different cultures. These reactions are reflected in the semantics of the studied verbs

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

In the framework of the projective geometric theory of systems of differential equations, which is being developed by the authors, conditions which ensure that a family of graphs of solutions of a system of m second-order ordinary differential equations y→̈ = f→(t, y→, y→̇) with m unknown functions y1(t), ⋯ , ym(t) can be straightened (that is, transformed into a family of straight lines) by means of a local diffeomorphism of the variables of the system which takes it to the form z→″ 00 = 0 (straightens the system) are investigated. It is shown that the system to be straightened must be cubic with respect to the derivatives of the unknown functions. Necessary and sufficient conditions for straightening the system are found, which have the form of differential equations for the coefficients of the system or are stated in terms of symmetries of the system. For m = 1 the system consists of a single equation ÿ = f→(t, y, ẏ), and the tests obtained reduce to the conditions for straightening this equations which were derived by Lie in 1883. © 2010 RAS(DoM) and LMS

Geometric theory of differential systems: Linearization criterion for systems of second-order ordinary differential equations with a 4-dimensional solvable symmetry group of the Lie-Petrov type VI 1

In the framework of projective-geometric theory of systems of differential equations developed by the authors, this paper studies the group properties of systems of two (resolved with respect to the second derivatives) second-order ordinary differential equations whose right-hand sides are polynomials of the third degree with respect to the derivatives of the unknown functions. A classification of such systems admitting four-dimensional symmetry group of the Lie-Petrov type VI 1 is given. For each of the systems, a necessary and sufficient linearization criterion is obtained, i.e., the authors find the necessary and sufficient conditions under which, by a change of variables, the system can be reduced to a differential system whose integral curves are straight lines and are expressed by three linear parametric equations or two linear equations with constant coefficients. For all linearizable systems, the linearizing changes of variables are indicated. © 2009 Springer Science+Business Media, Inc

Constructing a Space from the System of Geodesic Equations

Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from the Christoffel symbols. The procedure is extended for determining if a second order quadratically semi-linear system can be expressed as a system of geodesic equations, provided it has terms only quadratic in the first derivative apart from the second derivative term. A computer code has been developed for dealing with larger systems of geodesic equations

On Quantization of Polynomial Momentum Observables

The paper is devoted to quantization of polynomial momentum observables in the cotangent bundle of a smooth manifold. A quantization procedure is proposed allowing to quantize a wide class of functions which are polynomials of any order in momenta. In the last part of the paper the quantum mechanics of scalar particle in curved space-time is studied with the use of proposed approach.Comment: LaTeX 2.09, 8

Projective geometry of systems of second-order differential equations

It is proved that every projective connection on an n-dimensional manifold M is locally defined by a system script capital L sign of n - 1 second-order ordinary differential equations resolved with respect to the second derivatives and with right-hand sides cubic in the first derivatives, and that every differential system script capital L sign defines a projective connection on M. The notion of equivalent differential systems is introduced and necessary and sufficient conditions are found for a system y to be reducible by a change of variables to a system whose integral curves are straight lines. It is proved that the symmetry group of a differential system script capital L sign is a group of projective transformations in n-dimensional space with the associated projective connection and has dimension ≤ n 2 + 2n. Necessary and sufficient conditions are found for a system to admit the maximal symmetry group; basis vector fields and structure equations of the maximal symmetry Lie algebra are produced. As an application a classification is given of the systems script capital L sign of two second-order differential equations admitting three-dimensional soluble symmetry groups. © 2006 RAS(DoM) and LMS