Determination Of First Integrals Research Articles

A λ-symmetry-based method for the linearization and determination of first integrals of a family of second-order ordinary differential equations

We present a characterization of the second-order ordinary differential equations (ODEs) that can be linearized by means of certain nonlocal transformations. This characterization is given in terms of the coefficients of the equation and also determines the second-order ODEs that admit λ-symmetries and first integrals of some specific forms. Systematic methods to find these first integrals, λ-symmetries and the linearizing transformations are also derived.

Two-dimensional systems that arise from the Noether classification of Lagrangians on the line

Noether-like operators play an essential role in writing down the first integrals for Euler–Lagrange systems of ordinary differential equations (ODEs). The classification of such operators is carried out with the help of analytic continuation of Lagrangians on the line. We obtain the classification of 5, 6 and 9 Noether-like operators for two-dimensional Lagrangian systems that arise from the submaximal and maximal dimensional Noether point symmetry classification of Lagrangians on the line. Cases in which the Noether-like operators are also Noether point symmetries for the systems of two ODEs are mentioned. In particular, the 8-dimensional maximal Noether algebra is remarkably obtained for the simplest system of the free particle equations in two dimensions from the 5-dimensional complex Noether algebra of the standard Lagrangian of the scalar free particle equation. We present the effectiveness of Noether-like operators for the determination of first integrals of systems of two nonlinear differential equations which arise from scalar complex Euler–Lagrange ODEs that admit Noether symmetry.

SYMMETRIES AND FIRST INTEGRALS FOR NON-VARIATIONAL EQUATIONS

For a class of exterior ideals, we present a method associating first integrals of the characteristic distributions to symmetries of the ideal. The method is applied, under some assumptions, to the study of first integrals of ordinary differential equations and first order partial differential equations as well as to the determination of first integrals for integrable distributions of vector fields.

Approaches to solving nonlinear ODEs

Abstract Approaches for investigating nonlinear differential equations (DEs) are described and the status of corresponding programs is reported. Investigations include the determination of infinitesimal symmetries of single ODEs/PDEs or -systems, and for ODEs the determination of first integrals, of factorizations into DEs of lower order and of equivalent Lagrangians.

Investigating DEs with CRACK and related programs

The aim of the package to be described is to try modularizing the investigation of differential equations for which there are no complete algorithms available yet. All, that is available for such problems are algorithms for special situations, e.g. when first integrals with a simple structure exist (e.g. polynomial in first derivatives) or when the problem has infinitesimal symmetries. In all such cases, finally a system of differential equations has to be solved which is overdetermined in the sense that more conditions have to be satisfied than there are unknown functions. To do a variety of such investigations efficiently, like a symmetry analysis, application of symmetries, determination of first integrals, differential factors, equivalent Lagrangians, the strategy is to have one package (CRACK) for simplifying DEs and solving simple DEs as effective as possible and to use this program as the main tool for all the above mentioned investigations. For each investigation there is then only a short program necessary to just formulate the necessary conditions in form of an overdetermined DE-system and to call CRACK to solve this, possibly in a number of successive calls. The examples below shall indicate the range of possible applications.

First integrals via polynomial canonical transformations

Maharatna, Dutt, and Chattarji [J. Math. Phys. 20, 2221 (1979)] discussed the use of time-dependent canonical transformations for the determination of first integrals for time-dependent Hamiltonian systems. One particular proposal that successive time-dependent polynomial canonical transformations will enable first integrals to be found for a wider variety of time-dependent polynomial Hamiltonians than can be obtained using time-dependent linear canonical transformations is shown to be not true for the paradigm which they selected. It is suggested that their ansatz is ill-founded in general.