The approximate Noether symmetries and approximate first integrals for the approximate Hamiltonian systems

The authors prove that the standard least action principle implies a more general form of the same principle by which they can state generalized motion equation including the classical Euler equation as a particular case. This form is based on an observation regarding the last action principle about the limit case in the classical approach using symmetry violations. Furthermore the well known first integrals of the classical Euler equations become only approximate first integrals. The authors also prove a generalization of the fundamental lemma of the calculus of variation and we consider the application in electromagnetism.

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PERIODIC FIRST INTEGRALS FOR HAMILTONIAN SYSTEMS OF LIE TYPE

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887811005634 ◽

2011 ◽

Vol 08 (06) ◽

pp. 1169-1177 ◽

Cited By ~ 14

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.

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On First Integrals of Linear Hamiltonian Systems

Doklady Mathematics ◽

10.1134/s1064562418070256 ◽

2018 ◽

Vol 98 (3) ◽

pp. 616-618 ◽

Cited By ~ 1

Author(s):

A. B. Zheglov ◽

D. V. Osipov

Keyword(s):

Hamiltonian Systems ◽

First Integrals ◽

Linear Hamiltonian Systems

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First integrals of motion for two dimensional weight-homogeneous Hamiltonian systems in curved spaces

Communications in Nonlinear Science and Numerical Simulation ◽

10.1016/j.cnsns.2019.04.002 ◽

2019 ◽

Vol 75 ◽

pp. 220-235 ◽

Cited By ~ 3

Author(s):

A.A. Elmandouh

Keyword(s):

Hamiltonian Systems ◽

First Integrals ◽

Two Dimensional ◽

Integrals Of Motion ◽

Curved Spaces

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Approximate generalized symmetries, normal forms and approximate first integrals

Physics Letters A ◽

10.1016/s0375-9601(00)00021-9 ◽

2000 ◽

Vol 266 (2-3) ◽

pp. 106-122

Author(s):

G. Ünal

Keyword(s):

Normal Forms ◽

First Integrals ◽

Generalized Symmetries ◽

Approximate First Integrals

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A class of C∞-integrable Hamiltonian systems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500024161 ◽

1989 ◽

Vol 113 (3-4) ◽

pp. 293-314 ◽

Cited By ~ 3

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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