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

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.     

Algebraic properties of first integrals for systems of second-order ordinary differential equations of maximal symmetry

Symmetries of the first integrals for scalar linear or linearizable secondorder ordinary di?erential equations (ODEs) have already been derived and shown to exhibit interesting properties. One of these is that the symmetry algebra sl(3, IR) is generated by the three triplets of symmetries of the functionally independent first integrals and its quotient. In this paper, we first investigate the Lie-like operators of the basic first integrals for the linearizable maximally symmetric system of two second-order ODEs represented by the free particle system, obtainable from a complex scalar free particle equation, by splitting the corresponding complex basic first integrals and its quotient as well as their associated symmetries. It is proved that the 14 Lie-like operators corresponding to the complex split of the symmetries of the functionally independent first integrals I₁, I₂ and their quotient I₂/I₁ are precisely the Lie-like operators corresponding to the complex split of the symmetries of the scalar free particle equation in the complex domain. Then, it is shown that there are distinguished four symmetries of each of the four basic integrals and their quotients of the two-dimensional free particle system which constitute four-dimensional Lie algebras which are isomorphic to each other and generate the full symmetry algebra sl(4, IR) of the free particle system. It is further shown that the (n + 2)-dimensional algebras of the n + 2 first integrals of the system of n free particle equations are isomorphic to each other and generate the full symmetry algebra sl(n + 2, IR) of the free particle system.     

CANONICAL FORMS FOR PARTICLE LAGRANGIANS OF THE LINEAR SECOND-ORDER EQUATION

Abstract

A particle Lagrangian of a linear scalar second-order ordinary differential equation can admit maximally one of 1,2,3 or 5 Noether point symmetries. Moreover, canonical forms of particle Lagrangians of the linear equation are presented according to the number (and algebra) of Noether point symmetries they admit.     

The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations

A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan's equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for second-order Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.

     

INVARIANT VARIATIONAL PRINCIPLES INVOLVING VECTOR AND METRIC FIELDS IN 4-DIMENSIONS

Abstract

Variational principles in which the Lagrangian is a scalar density and a function of a metric tensor and a vector field, together with their first derivatives, are investigated in a 4-dimensional space. Associated with such Lagrangians are two expressions, the metric Euler-Lagrange expression and the vector Euler-Lagrange expression. The most general Lagrangians (of this kind) for which either of these Euler-Lagrange expressions vanishes identically, are obtained.

The most general Lagrangian (of this kind) for which the vector Euler-Lagrange equations are precisely Maxwell's equations is obtained. Although this Lagrangian is more general than the one commonly used, it still has essentially the same energy-momentum tensor.

The most general Lagrangian (of this kind) for which the metric Euler-Lagrange expression is precisely the electromagnetic energy-momentum tensor is derived. Although this Lagrangian is also more general than the one commonly used, the associated vector Euler-Lagrange equations are still Maxwell's equations.

Finally it is shown that, in contrast to the situation which obtains in the case of scalar densities which are functions of up to second derivatives of the metric and first derivatives of the vector field, there does not exist a Lagrangian, of the kind under investigation, for which the metric Euler-Lagrange expression is precisely the Einstein tensor and the vector Euler-Lagrange expression vanishes identically.     

Symmetry analysis and conservation laws of the geodesic equations for the Reissner-Nordström de Sitter black hole with a global monopole

This paper is devoted to the comprehensive analysis of the problem of symmetries and conservation laws for the geodesic equations of the Reissner-Nordström de Sitter (RNdS) black hole with a global monopole. For this purpose, the system of geodesic equations is determined and the corresponding classical Lie point symmetry operators are obtained. An optimal system of one dimensional subalgebras is constructed and a brief discussion about the algebraic structure of the Lie algebra of symmetries is presented. Also, the Noether symmetries of the geodesic Lagrangian is calculated. Finally, by applying two methods including Noether’s theorem and direct method the conservation laws associated to the system of geodesic equations are obtained.     

On the Jacobi Last Multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé–Gambier classification

We use a formula derived almost seventy years ago by Madhav Rao connecting the Jacobi Last Multiplier of a second-order ordinary differential equation and its Lagrangian and determine the Lagrangians of the Painlevé equations. Indeed this method yields the Lagrangians of many of the equations of the Painlevé–Gambier classification. Using the standard Legendre transformation we deduce the corresponding Hamiltonian functions. While such Hamiltonians are generally of non-standard form, they are found to be constants of motion. On the other hand for second-order equations of the Liénard class we employ a novel transformation to deduce their corresponding Lagrangians. We illustrate some particular cases and determine the conserved quantity (first integral) resulting from the associated Noetherian symmetry. Finally we consider a few systems of second-order ordinary differential equations and deduce their Lagrangians by exploiting again the relation between the Jacobi Last Multiplier and the Lagrangian.     

Approximate Noether-type symmetries and conservation laws via partial Lagrangians for PDEs with a small parameter

We show how one can construct approximate conservation laws of approximate Euler-type equations via approximate Noether-type symmetry operators associated with partial Lagrangians. The ideas of the procedure for a system of unperturbed partial differential equations are extended to a system of perturbed or approximate partial differential equations. These approximate Noether-type symmetry operators do not form a Lie algebra in general. The theory is applied to the perturbed linear and nonlinear (1+1) wave equations and the Maxwellian tails equation. We have also obtained new approximate conservation laws for these equations.     

The Hessian and Jacobi morphisms for higher order calculus of variations

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler-Lagrange morphism which turns out to be self-adjoint along solutions of the Euler-Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.     

Approximate conservation laws of nonlinear perturbed heat and wave equations

We construct approximate conservation laws for non-variational nonlinear perturbed (1+1) heat and wave equations by utilizing the partial Lagrangian approach. These perturbed nonlinear heat and wave equations arise in a number of important applications which are reviewed. Approximate symmetries of these have been obtained in the literature. Approximate partial Noether operators associated with a partial Lagrangian of the underlying perturbed heat and wave equations are derived herein. These approximate partial Noether operators are then used via the approximate version of the partial Noether theorem in the construction of approximate conservation laws of the underlying perturbed heat and wave equations.     

Symmetries, integrals and solutions of ordinary differential equations of maximal symmetry

Second- and third-order scalar ordinary differential equations of maximal symmetry in the traditional sense of point, respectively contact, symmetry are examined for the mappings they produce in solutions and fundamental first integrals. The properties of the ‘exceptional symmetries’, i.e. those not considered to be generic to scalar equations of maximal symmetry, can be recast into a form which is applicable to all such equations of maximal symmetry. Some properties of these symmetries are demonstrated.     

Noether’s Theorem for Constrained Hamiltonian System Under Generalized Operators北大核心CSCD

Noether’s symmetry and conserved quantity of singular systems under generalized operators were studied. Firstly, the Lagrangian equation of singular systems under generalized operators was established, and the primary constraints on the system were derived. Then the Lagrangian multiplier was introduced to establish the constrained Hamilton equation and the compatibility condition under generalized operators. Secondly, based on the invariance of the Hamilton action under the infinitesimal transformation, Noether’s theorem for constrained Hamiltonian systems under generalized operators was established, and the symmetry and corresponding conserved quantity of the system were given. Under certain conditions, Noether’s conservation of constrained Hamiltonian systems under generalized operators can be reduced to Noether’s conservation of integer-order constrained Hamiltonian systems. Finally, an example illustrates the application of the results. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Lie group classification of the generalized Lane–Emden equation

We carry out the Lie group classification of the generalized Lane–Emden equation xu^″+nu^′+xH(u)=0, which has many applications in mathematical physics and astrophysics. We show that the equation admits a three-dimensional equivalence Lie algebra. It is also shown that the principal Lie algebra, which in this case is trivial, has seven possible extensions. Three new cases arise for which the Lie point symmetry algebra is non-trivial. Comparison is then made of these cases with the Noether symmetry cases as well as the partial Noether operators.     

Conservation laws for a generalized coupled bidimensional Lane–Emden system

This paper aims to perform Noether symmetry classification of a generalized coupled bidimensional Lane–Emden system and computes the Noether operators corresponding to its first-order Lagrangian. In addition conservation laws of the various cases which admit Noether point symmetries are constructed for the underlying system.     

Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations

Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations, and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order (or systems) of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently, classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define a new type of transformation that builds upon this relation and obtain equivalence classes of scalar third order ODEs linearizable via these transformations. Four equivalence classes of such equations are seen to exist.     

Approximate partial Noether operators and first integrals for cubically coupled nonlinear Duffing oscillators subject to a periodically driven force

The approximate first integrals for a system of two cubically coupled nonlinear Duffing oscillators subject to a periodically driven force are constructed with the help of approximate partial Noether approach. Firstly, approximate partial Noether operators associated with a partial Lagrangian are derived. Then approximate first integrals are obtained for both the resonant and non-resonant cases.     

Classification of ordinary differential equations by conditional linearizability and symmetry

Lie’s invariant criteria for determining whether a second order scalar equation is linearizable by point transformation have been extended to third and fourth order scalar ordinary differential equations (ODEs). By differentiating the linearizable by point transformation scalar second order ODE (respectively third order ODE) and then requiring that the original equation holds, what is called conditional linearizability by point transformation of third and fourth order scalar ODEs, is discussed. The result is that the new higher order nonlinear ODE has only two arbitrary constants available in its solution. One can use the same procedure for the third and fourth order extensions mentioned above to get conditional linearizability by point or other types of transformation of higher order scalar equations. Again, the number of arbitrary constants available will be the order of the original ODE. A classification of ODEs according to conditional linearizability by transformation and classifiability by symmetry are proposed in this paper.     

Seeking (and finding) Lagrangians

It is well known that for any second-order ordinary differential equation (ODE), a Lagrangian always exists, and the key to its construction is the Jacobi last multiplier. Is it possible to find Lagrangians for first-order systems of ODEs or for higher-order ODEs? We show that the Jacobi last multiplier can also play a major role in this case.     

Higher-order supersymmetry in quantum mechanics and integrability of two-dimensional Hamiltonians

A new method of finding the dynamic symmetry operators in two-dimensional quantum systems is proposed. This method is based on the SUSY algebra with superchanges of higher order in derivatives. The symmetry operators arise when closing the SUSY algebra for a wide class of potentials, and, in some cases, they are of second order in derivatives. The solutions for potentials admitting symmetry operators of fourth order are also obtained. The investigation of the quasiclassical limit of the superalgebra results in new classical integrals of motion for a certain type of systems. Bibliography: 9 titles. In memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 68–80. Translated by V. A. Andrianov.