Rational first integrals of the Liénard equations: The solution to the Poincaré problem for the Liénard equations

The method of first integrals (MFI) based on the equation of motion for the displacement vector, or based on the one for the traction vector was introduced recently in order to find explicit secular equations of Rayleigh waves whose characteristic equations (i.e the equations determining the attenuation factor) are fully quartic or are of higher order (then the classical approach is not applicable). In this paper it is shown that, not only to Rayleigh waves, the MFI can be applicable also to other waves by running it on the equations for mixed vectors. In particular: (i) By applying the MFI to the equations for the displacement-traction vector we get the explicit dispersion equations of Stoneley waves in twinned crystals (ii) Running the MFI on the equations for the traction-electric induction vector and the traction-electrical potential vector provides the explicit dispersion equations of SH-waves in piezoelastic materials. The obtained dispersion equations are identical with the ones previously derived using the method of polarization vector, but the procedure of driving them is more simple.

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An Adomian decomposition method for solving Liénard equations in general form

ANZIAM Journal ◽

10.21914/anziamj.v51i0.398 ◽

2010 ◽

Vol 51 ◽

Author(s):

Mohammad Nili Ahmadabadi ◽

Farid Mohammad Maalek Ghaini

Keyword(s):

Decomposition Method ◽

Adomian Decomposition Method ◽

Liénard Equations ◽

Adomian Decomposition

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Poisson Brackets & Angular Momentum

10.1093/oso/9780198822370.003.0017 ◽

2018 ◽

Author(s):

Peter Mann

Keyword(s):

Generating Functions ◽

First Integrals ◽

Lagrangian Mechanics ◽

Poisson Brackets ◽

Coordinate Transformations ◽

Canonical Transformations ◽

Gauge Transformations ◽

The Third ◽

Point Transformations ◽

Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

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Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case

Journal of Applied Analysis ◽

10.1515/jaa-2018-0017 ◽

2018 ◽

Vol 24 (2) ◽

pp. 175-183

Author(s):

Jean-Claude Ndogmo

Keyword(s):

Explicit Form ◽

Nonlinear Equations ◽

Variational Principles ◽

Symmetry Algebra ◽

First Integrals ◽

Maximal Symmetry ◽

General Order ◽

Linear And Nonlinear ◽

Variational Symmetries ◽

Variational Symmetry

Abstract Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. A discussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations.

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