NON-NOETHER CONSERVATION LAWS

2012 ◽  
Vol 09 (04) ◽  
pp. 1220004
Author(s):  
PETRE BIRTEA ◽  
RĂZVAN M. TUDORAN
Keyword(s):  
Differential Operator ◽  
Hamiltonian System ◽  
Conservation Laws ◽  
Symplectic Form ◽  
Infinitesimal Symmetry

In this paper we will give a formula for computing conservation laws for a Hamiltonian system that admits non-Noether infinitesimal symmetry. The formula involves the differential operator associated with the dual Lefschetz operator corresponding to a symplectic form.

Lie symmetries analysis and conservation laws for the fractional Calogero–Degasperis–Ibragimov–Shabat equation

2018 ◽  
Vol 15 (07) ◽  
pp. 1850110 ◽  
Author(s):  
S. Sahoo ◽  
S. Saha Ray
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Operator ◽  
Conservation Laws ◽  
Symmetry Analysis ◽  
Lie Symmetries ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Conservation Theorem

The present paper includes the study of symmetry analysis and conservation laws of the time-fractional Calogero–Degasperis–Ibragimov–Shabat (CDIS) equation. The Erdélyi–Kober fractional differential operator has been used here for reduction of time fractional CDIS equation into fractional ordinary differential equation. Also, the new conservation theorem has been used for the analysis of the conservation laws. Furthermore, the new conserved vectors have been constructed for time fractional CDIS equation by means of the new conservation theorem with formal Lagrangian.

scholarly journals The shallow water equations: conservation laws and symplectic geometry

1987 ◽  
Vol 10 (3) ◽  
pp. 557-562 ◽  
Author(s):  
Yilmaz Akyildiz
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Symplectic Form ◽  
Symplectic Geometry ◽  
Nonlinear Differential Equations ◽  
Shallow Water Waves ◽  
Hodograph Method ◽  
Sloping Bottom

We consider the system of nonlinear differential equations governing shallow water waves over a uniform or sloping bottom. By using the hodograph method we construct solutions, conservation laws, and Böcklund transformations for these equations. We show that these constructions are canonical relative to a symplectic form introduced by Manin.

scholarly journals Symmetry properties of conservation laws

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640003 ◽  
Author(s):  
Stephen C. Anco
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Conservation Law ◽  
Conserved Quantity ◽  
Partial Differential ◽  
Standard Formula ◽  
Infinitesimal Symmetry ◽  
Symmetry Properties ◽  
General Method

Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law and its associated conserved quantity are invariant (and, more generally, homogeneous) under the action of a symmetry. These results are used to show that a recent conservation law formula (due to Ibragimov) is equivalent to a standard formula for the action of an infinitesimal symmetry on a conservation law multiplier.

Measure expansive symplectic diffeomorphisms and Hamiltonian systems

2016 ◽  
Vol 27 (09) ◽  
pp. 1650077
Author(s):  
Manseob Lee ◽  
Junmi Park
Keyword(s):  
Riemannian Manifold ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Symplectic Form ◽  
Symplectic Diffeomorphisms ◽  
Symplectic Diffeomorphism

Let [Formula: see text] be a [Formula: see text]-dimensional ([Formula: see text]), compact smooth Riemannian manifold endowed with a symplectic form [Formula: see text]. In this paper, we show that, if a symplectic diffeomorphism [Formula: see text] is [Formula: see text]-robustly measure expansive, then it is Anosov and a [Formula: see text] generic measure expansive symplectic diffeomorphism [Formula: see text] is mixing Anosov. Moreover, for a Hamiltonian systems, if a Hamiltonian system [Formula: see text] is robustly measure expansive, then [Formula: see text] is Anosov.

scholarly journals Legendre Polynomials Spectral Approximation for the Infinite-Dimensional Hamiltonian Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Zhongquan Lv ◽  
Mei Xue ◽  
Yushun Wang
Keyword(s):  
Differential Operator ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Legendre Polynomials ◽  
Hamiltonian Operator ◽  
Spectral Approximation ◽  
Infinite Dimensional

This paper considers a Legendre polynomials spectral approximation for the infinite-dimensional Hamiltonian systems. As a consequence, the Legendre polynomials spectral semidiscrete system is a Hamiltonian system for the Hamiltonian system whose Hamiltonian operator is a constant differential operator.

Quasilinear Hamiltonian systems

1987 ◽  
Vol 106 (3-4) ◽  
pp. 195-204
Author(s):  
Jan S. Rogulski
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Symplectic Form ◽  
Scalar Product ◽  
Sufficient Conditions ◽  
Hamiltonian Form ◽  
Quasilinear Systems ◽  
Necessary And Sufficient ◽  
Evolution Systems ◽  
General Method

SynopsisWe consider quasilinear systems of 2N partial differential equations with 2N unknown functions depending on n + 1 variables as evolution systems on the space L2(Rn, RN) × L2(Rns, RN) endowed with a symplectic form induced by the standard scalar product on L2(Rn, RN). The necessary and sufficient conditions for such a system to be a Hamiltonian system are derived. The main purpose of this paper is to propose a straightforward link between the symplectic approach formulated by Chernoff, Hughes and Marsden and the multisymplectic formulations of evolution systems created by Kijowski and developed by Gawedzki and Kondracki. A general method of constructing the multisymplectic form and the Hamiltonian form for these systems is given.

scholarly journals Conservation Laws for a Delayed Hamiltonian System in a Time Scales Version

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 668 ◽  
Author(s):  
Xiang-Hua Zhai ◽  
Yi Zhang
Keyword(s):  
Hamiltonian System ◽  
Conservation Laws ◽  
Time Scales ◽  
Scientific Research ◽  
Noether Theorem ◽  
Continuous Version ◽  
Difference Analysis ◽  
Delayed Arguments ◽  
New Perspective ◽  
Fowler Equation

The theory of time scales which unifies differential and difference analysis provides a new perspective for scientific research. In this paper, we derive the canonical equations of a delayed Hamiltonian system in a time scales version and prove the Noether theorem by using the method of reparameterization with time. The results extend not only the continuous version of the Noether theorem with delayed arguments but also the discrete one. As an application of the results, we find a Noether-type conserved quantity of a delayed Emden-Fowler equation on time scales.

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