First Integrals Of Dynamical Systems Research Articles

Transcendental First Integrals of Dynamical Systems on the Tangent Bundle to the Sphere

In this paper, we examine the existence of transcendental first integrals for some classes of systems with symmetries. We obtain sufficient conditions of existence of first integrals of second-order nonautonomous homogeneous systems that are transcendental functions (in the sense of the theory of elementary functions and in the sense of complex analysis) expressed as finite combinations of elementary functions.

A partial Lagrangian method for dynamical systems

We develop a new approach termed as a discount free or partial Lagrangian method for construction of first integrals for dynamical systems of ordinary differential equations (ODEs). It is shown how one can utilize the Legendre transformation in a more general setting to provide the equivalence between a current value Hamiltonian and a partial or discount free Lagrangian when it exists. As a consequence, we develop a discount factor free Lagrangian framework to deduce reductions and closed-form solutions via first integrals for ODEs arising from economics by proving three important propositions. The approach is algorithmic and applies to many state variables of the Lagrangian. In order to show its effectiveness, we apply the method to models, one linear and two nonlinear, with one state variable. We obtain new exact solutions for the last model. The discount free Lagrangian naturally arises in economic growth theory and many other economic models when the control variables can be eliminated at the outset which is not always possible in optimal control theory applications of economics. We explain our method with the help of few widely used economic growth models. We point out the difference between this approach and the more general partial Hamiltonian method proposed earlier for a current value Hamiltonian (Naz et al. in Commun Nonlinear Sci Numer Simul 19:3600–3610, 2014) which is applicable in a general setting involving time, state, costate and control variables.

Determining Liouvillian first integrals for dynamical systems in the plane and an integrability analysis

Determining Liouvillian first integrals for dynamical systems in the plane and an integrability analysis

Determining Liouvillian first integrals for dynamical systems in the plane

Here we present/implement a semi-algorithm to find Liouvillian first integrals of dynamical systems in the plane. The algorithm is based on a Darboux-type procedure to find the integrating factor for the system. Since the particular form of such systems allows reducing it to a single rational first order ordinary differential equation (rational first order ODE), the Lsolver package presents a set of software routines in Maple for dealing with rational first order ODEs. The package present commands permitting research investigations of some algebraic properties of the system that is being studied. Program summary Program title:Lsolver Catalogue number:ADZF_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZF_v1_0.html Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.:1513 No. of bytes in distributed program, including test data, etc.:17 367 Distribution format:tar.gz Programming language:Maple Computer:Any computer running Maple Operating system:Windows ME, Windows XP Classification:4.3 Nature of problem: Solution of rational first order ordinary differential equations. Solution method:The method of solution is based on a Darboux/PS type approach. Restrictions:If the integrating factor for the ODE under consideration presents Darboux Polynomials of high degree (>3) in the dependent and independent variables, the package may spend an impractical amount of time to obtain the solution. Unusual features:Our implementation not only searches for Liouvillian conserved quantities, but can also be used as a research tool that allows the user to follow all the steps of the procedure (for example, we can calculate the algebraic invariants curves and associated co-factors, the integrating factor etc). In addition, since our package is based on recent theoretical developments [J. Avellar, L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Integrating first-order differential equations with Liouvillian solutions via quadratures: a semi-algorithmic method, J. Comput. Appl. Math. 182 (2005) 327–332], it can successfully solve a class of rational first order ODEs that were not solved by some of the best-known ODE solvers available. Running time:This depends strongly on the ODE, but usually under 4 seconds.

A geometrical method towards first integrals for dynamical systems

We develop a method, based on Darboux’s and Liouville’s works, to find first integrals and/or invariant manifolds for a physically relevant class of dynamical systems, without making any assumption on these elements’ forms. We apply it to three dynamical systems: Lotka–Volterra, Lorenz and Rikitake.

Investigation of Painlevé property under time singularities transformations

In this paper, the relation between the Painlevé property for ordinary differential equations and some coordinate transformations in the complex plane is investigated. Two sets of transformations are introduced for which the transformations of singularities structure in the complex plane are explicitly computed. The first set acts only on the dependent variables while the second one acts also on the independent variables. It allows for defining an extended Painlevé test which is coordinates invariant. Our results are applied on various problems arising in the singularity analysis such as the problem of negative resonances and the weak Painlevé conjecture. The method for finding new first integrals for dynamical systems is also shown.

SOME CONSTRUCTIONS OF INTEGRABLE DYNAMICAL SYSTEMS

New constructions of integrable dynamical systems are found that admit representation as Lax matrix equations. A countable set of integrable systems is constructed which in the continuous limit turn into the Korteweg-de Vries equation. For an arbitrary space with finite measure and measure-preserving mapping differential equations are constructed on the space of measurable functions on . Here differentiation is with respect to time and the equations have a countable set of first integrals. Constructions are also given for first integrals of dynamical systems preserving certain differential forms, and new constructions of matrix differential equations having large families of first integrals. Bibliography: 18 titles.

Another method for the obtention of first integrals of dynamical systems

A computational procedure is developed leading to the obtention of first integrals of a dynamical systemX under the assumption thatX belongs to a certain involutive distributionD and that an external symmetrySofD is known.

On symmetries and first integrals of dynamical systems

A previous result according to which, given a symmetry vector of a dynamical system, under certain co-ordinate-dependent conditions, a first integral could be expressed as the divergence of the symmetry vector has been generalized. By introducing convenient connections it is found that not only one but a set of first integrals can, in general, be interpreted as divergences of a single symmetry vector. It is also shown that any symmetry vector is a linear combination of a maximal independent set of symmetry vectors of the same dynamical system with first integrals as coefficients. This leads to other relationships between first integrals andsets of symmetry vectors. Finally it is seen how a type-(r, s) tensor invariant of the one-parameter group generated by a dynamical system yieldsr generally different type-(r−1),s) tensor invariants. This is, in a sense, a further generalization of the former result. The type-(r, s) tensor invariant also leads to a type-(r, s+1) tensor invariant of the same dynamical system.

Symmetry vectors and first integrals of dynamical systems

Symmetry vectors and first integrals of dynamical systems

Notes on the connection between the symmetries and the first integrals of dynamical systems

Notes on the connection between the symmetries and the first integrals of dynamical systems

The related integral theorem in phase space: A generalization of Poisson’ theorem on constants of the motion

The related integral theorem 0 -3) is a unified method which shows that , if a dynamical system admits a symmetry, then in general a new constant of the motion will result from the deformation of a given one under this symmetry mapping. As originally formulated 0) the theorem provided a method for generating quadrat ic first integrals of dynamical systems with geodesic trajectories based upon the deformation of the metrical quadrat ic integral under projective collineations. A later generalization (3) extended the domain of the theorem to include conservative dynamical systems governed by the equation D2x~/dt2+ gV.~= 0(~). Dynamical symmetries for such systems were called t ra jectory collineations. Based upon the deformations of the energy integral under the t ra jec tory collineation mappings addi t ional constants of the motion were obtained in a systematic manner. Application of the theory to the Kepler problem and three-dimensional isotropic harmonic oscillator showed that the well-known RungeLenz vector constant of the motion and symmetr ic tensor constant of the motion could be obtained in a simple manner by this unified approach. In the cases thus far considered the related integral theorem has been formulated in terms of point mappings of the generalized co-ordinates of configuration space. We now show how the related integral theorem may be formulated in phase space for dynamical systems governed by Hamil ton 's equations. As a result we find tha t the wellknown Poisson's theorem on constants of the inotion (b) is a special caae of the related integral theorem. Consider a dynamical system of n degrees of freedom with Hamiltonian H(~). I Iamil ton 's equations may be expressed in the form (s e)

A group-variational procedure for finding first integrals of dynamical systems

In this work a group-variational procedure is given for finding first integrals of non-conservative dynamical systems. The existence of first integrals depends on the existence of solutions of the system of partial differential equations—generalized Killing's equations. Two examples are analyzed in detail.