scholarly journals First integrals of holonomic systems without Noether symmetries

2020 ◽  
Vol 61 (12) ◽  
pp. 122701
Author(s):  
Michael Tsamparlis ◽  
Antonios Mitsopoulos
Keyword(s):  
First Integrals ◽  
Noether Symmetries ◽  
Holonomic Systems

scholarly journals A study of the approximate singular Lagrangian-conditional Noether symmetries and first integrals

2019 ◽  
Vol 16 (03) ◽  
pp. 1950033 ◽  
Author(s):  
Sameerah Jamal
Keyword(s):  
Degrees Of Freedom ◽  
Constraint Equation ◽  
First Integrals ◽  
Conformal Killing ◽  
Noether Symmetries ◽  
Geometric Framework ◽  
Full Generality ◽  
Conformal Killing Vectors ◽  
Gauge Fixing ◽  
Symmetry Generators

The investigation of approximate symmetries of reparametrization invariant Lagrangians of [Formula: see text] degrees of freedom and quadratic velocities is presented. We show that extra conditions emerge which give rise to approximate and conditional Noether symmetries of such constrained actions. The Noether symmetries are the simultaneous conformal Killing vectors of both the kinetic metric and the potential. In order to recover these conditional symmetry generators which would otherwise be lost in gauge fixing the lapse function entering the perturbative Lagrangian, one must consider the lapse among the degrees of freedom. We establish a geometric framework in full generality to determine the admitted Noether symmetries. Additionally, we obtain the corresponding first integrals (modulo a constraint equation). For completeness, we present a pedagogical application of our method.

scholarly journals F(R,G) Cosmology through Noether Symmetry Approach

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 719 ◽  
Author(s):  
Ugur Camci
Keyword(s):  
Cosmological Solution ◽  
First Integrals ◽  
Boundary Term ◽  
Noether Symmetry ◽  
Noether Symmetries ◽  
Real Numbers ◽  
Cosmological Scale ◽  
Symmetry Approach ◽  
Scale Factors ◽  
Functional Boundary

The F ( R , G ) theory of gravity, where R is the Ricci scalar and G is the Gauss-Bonnet invariant, is studied in the context of existence the Noether symmetries. The Noether symmetries of the point-like Lagrangian of F ( R , G ) gravity for the spatially flat Friedmann-Lemaitre-Robertson-Walker cosmological model is investigated. With the help of several explicit forms of the F ( R , G ) function it is shown how the construction of a cosmological solution is carried out via the classical Noether symmetry approach that includes a functional boundary term. After choosing the form of the F ( R , G ) function such as the case ( i ) : F ( R , G ) = f 0 R n + g 0 G m and the case ( i i ) : F ( R , G ) = f 0 R n G m , where n and m are real numbers, we explicitly compute the Noether symmetries in the vacuum and the non-vacuum cases if symmetries exist. The first integrals for the obtained Noether symmetries allow to find out exact solutions for the cosmological scale factor in the cases (i) and (ii). We find several new specific cosmological scale factors in the presence of the first integrals. It is shown that the existence of the Noether symmetries with a functional boundary term is a criterion to select some suitable forms of F ( R , G ) . In the non-vacuum case, we also obtain some extra Noether symmetries admitting the equation of state parameters w ≡ p / ρ such as w = − 1 , − 2 / 3 , 0 , 1 etc.

Emden-Fowler type system: noether symmetries and first integrals

2012 ◽  
Vol 32 (5) ◽  
pp. 1959-1966 ◽  
Author(s):  
B. Muatjetjeja ◽  
C.M. Khalique
Keyword(s):  
Type System ◽  
First Integrals ◽  
Noether Symmetries

scholarly journals Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1180
Author(s):  
Muhammad Safdar ◽  
Asghar Qadir ◽  
Muhammad Umar Farooq
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
First Integrals ◽  
Second Order ◽  
Complex Method ◽  
Two Dimensional ◽  
Noether Symmetries ◽  
The Real ◽  
Complex Methods

Noether symmetries and first integrals of a class of two-dimensional systems of second order ordinary differential equations (ODEs) are investigated using real and complex methods. We show that first integrals of systems of two second order ODEs derived by the complex Noether approach cannot be obtained by the real methods. Furthermore, it is proved that a complex method can be extended to larger systems and higher order.

THE DYNAMICAL NOETHER SYMMETRIES OF A BOSONIC q-OSCILLATOR

1999 ◽  
Vol 14 (22) ◽  
pp. 3543-3563
Author(s):  
TIAGO J. M. SIMÕES
Keyword(s):  
Phase Space ◽  
Classical Model ◽  
Small Deformation ◽  
First Integrals ◽  
Noether Symmetries ◽  
Oscillator System ◽  
Dynamical Phase ◽  
Deformation Limit ◽  
Highly Nonlinear ◽  
Dynamical Algebra

The classical dynamical (phase space) Noether symmetries which correspond to the quantum, one-dimensional, bosonic, deformed "Biedenharn–Macfarlane q-oscillator" as defined by V. I. Man'ko and others, are given for small values of the parameter q by considering the model as a nondeformed theory with a highly nonlinear but of small strength interaction. For this nonconstrained one-degree of freedom system and by applying Noether's procedure in the form established by Katzin and Levine for velocity dependent transformations, we found the corresponding two functionally independent phase space first integrals. These classical integrals, as we explicitly prove, lead to a finitely generated infinite Poisson bracket dynamical algebra of first integrals which generalizes a recently obtained Noether dynamical algebra of the nondeformed harmonic oscillator system. We also show that a subalgebra of that infinite dynamical algebra, after quantization of the small-q classical model here proposed, corresponds exactly to the small deformation limit of the deformed quantum spectrum generating algebra su q(1,1) previously obtained for the q-oscillator system, on purely quantum grounds, by Kulish and Damaskinsky.

Method of first integrals and interface, surface waves

2010 ◽  
Vol 32 (2) ◽  
pp. 107-120
Author(s):  
Pham Chi Vinh ◽  
Trinh Thi Thanh Hue ◽  
Dinh Van Quang ◽  
Nguyen Thi Khanh Linh ◽  
Nguyen Thi Nam
Keyword(s):  
Rayleigh Waves ◽  
Displacement Vector ◽  
Attenuation Factor ◽  
Electrical Potential ◽  
Polarization Vector ◽  
First Integrals ◽  
Dispersion Equations ◽  
Interface Surface ◽  
Electric Induction ◽  
The One

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.

Poisson Brackets & Angular Momentum

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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