Darboux Polynomials and First Integrals of Polynomial Hamiltonian Systems

2021 ◽  
Author(s):  
Andrei Pranevich ◽  
Alexander Grin ◽  
Eduard Musafirov
Keyword(s):  
Hamiltonian Systems ◽  
First Integrals ◽  
Darboux Polynomials

First integrals and Darboux polynomials of natural polynomial Hamiltonian systems

2010 ◽  
Vol 374 (47) ◽  
pp. 4746-4748 ◽  
Author(s):  
Isaac A. García ◽  
Maite Grau ◽  
Jaume Llibre
Keyword(s):  
Hamiltonian Systems ◽  
First Integrals ◽  
Darboux Polynomials

scholarly journals Darboux polynomials and first integrals of natural polynomial Hamiltonian systems

2004 ◽  
Vol 326 (3-4) ◽  
pp. 219-226 ◽  
Author(s):  
Andrzej J. Maciejewski ◽  
Maria Przybylska
Keyword(s):  
Hamiltonian Systems ◽  
First Integrals ◽  
Darboux Polynomials

scholarly journals On the Integrability of a Tritrophic Food Chain Model

2010 ◽  
Vol 10 (2) ◽  
Author(s):  
Jaume Llibre ◽  
Clàudia Valls
Keyword(s):  
Food Chain ◽  
Formal Series ◽  
Complete Characterization ◽  
First Integrals ◽  
Chain Model ◽  
Darboux Polynomials ◽  
Analytic First Integrals ◽  
Exponential Factors ◽  
Food Chain Model

AbstractIn this paper we work with a vastly analyzed tritrophic food chain model. We provide a complete characterization of their Darboux polynomials and of their exponential factors. We also show the non-existence of polynomial first integrals, of rational first integrals, of local analytic first integrals in a neighborhood of the origin, of first integrals that can be described by formal series and of Darboux first integrals.

scholarly journals PERIODIC FIRST INTEGRALS FOR HAMILTONIAN SYSTEMS OF LIE TYPE

2011 ◽  
Vol 08 (06) ◽  
pp. 1169-1177 ◽  
Author(s):  
RUBEN FLORES ESPINOZA
Keyword(s):  
Hamiltonian Systems ◽  
Nonlinear Oscillator ◽  
Original System ◽  
First Integrals ◽  
Time Dependent ◽  
Existence Problem ◽  
Adjoint Group ◽  
Killing Form ◽  
Modern Physics ◽  
Existence Of Periodic Solutions

In this paper, we study the existence problem of periodic first integrals for periodic Hamiltonian systems of Lie type. From a natural ansatz for time-dependent first integrals, we refer their existence to the existence of periodic solutions for a periodic Euler equation on the Lie algebra associated to the original system. Under different criteria based on properties for the Killing form or on exponential properties for the adjoint group, we prove the existence of Poisson algebras of periodic first integrals for the class of Hamiltonian systems considered. We include an application for a nonlinear oscillator having relevance in some modern physics applications.

On First Integrals of Linear Hamiltonian Systems

2018 ◽  
Vol 98 (3) ◽  
pp. 616-618 ◽  
Author(s):  
A. B. Zheglov ◽  
D. V. Osipov
Keyword(s):  
Hamiltonian Systems ◽  
First Integrals ◽  
Linear Hamiltonian Systems

scholarly journals On Darboux polynomials and rational first integrals of the generalized Lorenz system

2014 ◽  
Vol 138 (3) ◽  
pp. 317-322 ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Lorenz System ◽  
First Integrals ◽  
Darboux Polynomials ◽  
Generalized Lorenz System

A class of C∞-integrable Hamiltonian systems

1989 ◽  
Vol 113 (3-4) ◽  
pp. 293-314 ◽  
Author(s):  
W. M. Oliva ◽  
M. S. A. C. Castilla
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
First Integrals ◽  
Integrable Hamiltonian Systems ◽  
Integral Of Motion ◽  
Toda System ◽  
Nonnegative Coefficients ◽  
Special Case ◽  
Three Degrees Of Freedom

SynopsisWe discuss the C∞ complete integrability of Hamiltonian systems of type q = —grad V(q) = F(q), in which the closure of the cone generated (with nonnegative coefficients) by the vectors F(q), q ϵ ℝn, does not contain a line. The components of the asymptotic velocities are first integrals and the main aim is to prove their smoothness as functions of the initial conditions. The Toda-like system with potential V(q)=ΣNi=1 exp(fi∣ q) is a special case of the considered systems ifthe cone C(f1,…,fN)={ΣNi=1cifi,ci≧0} does notcontain a line. In any number of degrees of freedom, if C(f1,…,fN) has amplitude not too large (ang (fi, fj ≦π/2i,j=1,2,…, N), the first integrals are C∞ functions. In two degrees of freedom, without restriction on the amplitude of the cone, C∞-integrability is proved even in a case in which it is known that there is no other meromorphic integral of motion independent of energy. In three degrees of freedom the C∞-integrability of a deformation of the classic nonperiodic Toda system is proved. Some other examples are also discussed.

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