Classical Noether Theorem Research Articles

Computational approach in obtaining analytic solutions of a generalized nonlinear breaking soliton equation with applications in engineering and physics

Higher-order nonlinear wave models have been a source of attraction to a huge number of researchers in recent times as a result of their significance in mathematical physics, other nonlinear sciences as well as engineering. In consequence, we outline in this paper the analytical studies entrenched on a generalized structure of a nonlinear breaking soliton equation with higher-order nonlinearity in four variables which have applications in science as well as engineering. Lie group theory is utilized to generate an 11-dimensional Lie algebra associated with the equation under consideration and in addition one parametric group of transformations related to the algebra is calculated. Besides, the technique is further invoked in performing reductions of the various subalgebras of the understudy model. Moreover, in conjunction with the theory, the direct integration technique is engaged to secure an analytic solution of the equation and as a result, a general analytic solution with regard to the second-kind elliptic-integral function is furnished. Moreover, we engage the novel simplest equation technique to gain more general solutions to the equation. In consequence, solitonic solutions comprising periodic, dark-bright, topological kink as well as singular solutions are achieved. We complemented that by exhibiting the dynamics of the secured solutions with the aid of graphical representations. In conclusion, we calculate conserved quantities associated with the aforementioned equation by invoking the well-celebrated classical Noether theorem technique.

Lie symmetry analysis, particular solutions and conservation laws for the dissipative (2 + 1)- dimensional AKNS equation

<abstract><p>The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.</p></abstract>

Noether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systems

In the present work, we formulate a generalization of the Noether Theorem for action-dependent Lagrangian functions. The Noether's theorem is one of the most important theorems for physics. It is well known that all conservation laws, \textrm{e.g.}, conservation of energy and momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether theorem cannot be applied to study non-conservative systems because it is not possible to formulate physically meaningful Lagrangian functions for this kind of systems in the classical calculus of variation. On the other hand, recently it was shown that an Action Principle with action-dependent Lagrangian functions provides physically meaningful Lagrangian functions for a huge variety of non-conservative systems (classical and quantum). Consequently, the generalized Noether Theorem we present enable us to investigate conservation laws of non-conservative systems. In order to illustrate the potential of application, we consider three examples of dissipative systems and we analyze the conservation laws related to spacetime transformations and internal symmetries.

New exact spherically symmetric solutions in f(R,ϕ,X) gravity by Noether's symmetry approach

The exact solutions of spherically symmetric space-times are explored by using Noether symmetries in f(R,ϕ,X) gravity with R the scalar curvature, ϕ a scalar field and X the kinetic term of ϕ. Some of these solutions can represent new black holes solutions in this extended theory of gravity. The classical Noether approach is particularly applied to acquire the Noether symmetry in f(R,ϕ,X) gravity. Under the classical Noether theorem, it is shown that the Noether symmetry in f(R,ϕ,X) gravity yields the solvable first integral of motion. With the conservation relation obtained from the Noether symmetry, the exact solutions for the field equations can be found. The most important result in this paper is that, without assuming R=constant, we have found new spherically symmetric solutions in different theories such as: power-law f(R)=f0 Rn gravity, non-minimally coupling models between the scalar field and the Ricci scalar f(R,ϕ,X)=f0 Rn ϕm+f1 Xq−V(ϕ), non-minimally couplings between the scalar field and a kinetic term f(R,ϕ,X)=f0 Rn +f1ϕmXq , and also in extended Brans-Dicke gravity f(R,ϕ,X)=U(ϕ,X)R. It is also demonstrated that the approach with Noether symmetries can be regarded as a selection rule to determine the potential V(ϕ) for ϕ, included in some class of the theories of f(R,ϕ,X) gravity.

Path-independent integrals in electrochemomechanical systems with flexoelectricity

Recently, a larger flexoelectric-like response in a bending semiconductor was found by Narvaez et al. This fact indicates that the electrochemical processes have a great influence on flexoelectricity. Thus, study on the coupling between the electrochemical process and the flexoelectric effect is crucial. In this paper, the classical Noether theorem is applied to the dissipative electrochemomechanical processes with consideration of the flexoelectric effect and the strain gradient. The necessary and sufficient conditions for obtaining the symmetry transformation are derived. Several conservation integrals, including J- and L-integrals, are obtained. Then, the relation of the J- and L-integrals with the energy release rate is verified. We also show that J-integral here can reduce to the result in previous work. Finally, a Mode III crack problem taking the flexoelectric effect into account is discussed to show the path-independence of the J-integral.

Scaling symmetries, conservation laws and action principles in one-dimensional gas dynamics

Scaling symmetries of the planar, one-dimensional gas dynamic equations with adiabatic index γ are used to obtain Lagrangian and Eulerian conservation laws associated with the symmetries. The known Eulerian symmetry operators for the scaling symmetries are converted to the Lagrangian form, in which the Eulerian spatial position of the fluid element is given in terms of the Lagrangian fluid labels. Conditions for a linear combination of the three scaling symmetries to be a divergence or variational symmetry of the action are established. The corresponding Lagrangian and Eulerian form of the conservation laws are determined by application of Noether's theorem. A nonlocal conservation law associated with the scaling symmetries is obtained by applying a nonlocal symmetry operator to the scaling symmetry-conserved vector. An action principle incorporating known conservation laws using Lagrangian constraints is developed. Noether's theorem for the constrained action principle gives the same formulas for the conserved vector as the classical Noether theorem, except that the Lie symmetry vector field now includes the effects of nonlocal potentials. Noether's theorem for the constrained action principle is used to obtain nonlocal conservation laws. The scaling symmetry conservation laws only apply for special forms of the entropy of the gas.

A Methodology for Finding Invariants of Motion for Asymmetric Systems with Gauge-Transformed Umbra Lagrangian Generated by Bond Graphs

The purpose of this article is to obtain conservation laws (invariants of motion) for different energy domains through the extended Noether theorem and bond graphs. Bond graphs are profitably used in representing the physics of a system as well as obtaining its umbra-Lagrangian. The article extends Lagrangian-Hamiltonian mechanics to deal with asymmetries in the system, which incorporates dissipative and nonpotential fields in a compact Lagrangian form, such that one may obtain invariants of motion through extension of Noether’s theorem. A detailed methodology is outlined in this article for obtaining the invariants of motion for a general class of asymmetric systems with a gauge-transformed umbra-Lagrangian. Symmetrization of an asymmetric system is introduced through the concept of gauge functions, for which the classical Noether theorem is extended over vector fields in the extended manifold comprising real and umbra displacements and velocities, as well as real time. A generalization of the variational principle or least action principle is also presented, which leads to the proposed form of the umbra-Lagrange equation through recursive minimization of functionals. Several illustrative examples are given to elucidate this concept in different physical contexts.

Second Noether-type theorem for the generalized variational principle of Herglotz

The generalized variational principle of Herglotz defines the functional, whose extrema are sought, by a differential equation rather than by an integral. For such functionals the classical Noether theorems are not applicable. First and second Noether-type theorems which do apply to the generalized variational principle of Herglotz were formulated and proved. These theorems contain the classical first and second Noether theorems as special cases. We published the first Noether-type theorem previously in this journal. Here we prove the second Noether-type theorem and show that it reduces to the classical second Noether theorem when the Herglotz variational principle reduces to the classical variational principle.

Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations

For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations. Results are also obtained for variational problems with higher-order derivatives.

A Proper Extension of Noether's Symmetry Theorem for Nonsmooth Extremals of the Calculus of Variations

For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations

On the conserved quantities and associated symmetries for some classes of perturbed wave equations with non-linearities

We show that the conserved vectors for a large class of the perturbed equations u tt − u xx + g( u)= εf( x, t, u, u x , u t ) (e.g., Klein–Gordon and sine-Gordon equations) can be determined using some recent results involving symmetry-based methods which do not require a knowledge of a Lagrangian. However, when a Lagrangian of the perturbed equation is known, we demonstrate that conservation laws associated with symmetries can still be found using an extended (approximate) form of the classical Noether's theorem.

Symmetries, conservation laws and variational principles for vector field theories†

The interplay between symmetries, conservation laws, and variational principles is a rich and varied one and extends well beyond the classical Noether's theorem. Recall that Noether's first theorem asserts that to every r dimensional Lie algebra of (generalized) symmetries of a variational problem there are r conserved quantities for the corresponding Euler-Lagrange equations. Noether's second theorem asserts that infinite dimensional symmetry algebras (depending upon arbitrary functions of all the independent variables) lead to differential identities for the Euler-Lagrange equations.

Noether Theorem and Its Inverse Theorem in Canonical Formalism for Nonholonomic Nonconservative Singular System

Classical Noether theorem and its generalization are given in configuration space and expressed in terms of Lagrange variables. The Noether Theorems 1, 2 in phase space for holonomic singluar system (of which the Lagrangian function is singular) are discussed, Here the Noether theorem and its inverse theorem in canonical formalism for nonholonomic nonconservative singular system are further investigated. Consider a nonholonomic nonconservative system whose motion is described in terms

Quantum basic invariants and classical Noether's theorem

Taking the generalized time-dependent harmonic oscillator (GTHO) and singular time-dependent harmonic oscillator (STHO) as illustrative examples, we discuss the connection between the quantum basic invariants and classical Noether's theorem and, for the system, find: (1) The classical basic invariants and general form of classical invariants. (2) The current algebra and corresponding operator algebra. (3) A kind of converse of the Noether's theorem. (4) The connection between the so-called symmetry algebra ({Gk} algebra) and the classical current algebra through the operator algebra.

A New Approach to the Study of Noether Invariants and Symmetry Algebras

In this paper, the driven generalized time-dependent harmonic oscillator (DGTHO) is studied by making use of the new concept of quantum basic invariants. The connection between the quantum basic invariants and classical Noether's theorem is discussed. The classical basic invariants as well as the general form of classical invariants for the system are found. From these, we get the five Noether invariants and the set of corresponding infinitesimal generators {G(k)}, hence, generalizing the work of G. Profilo and G. Soliani. Also, for the system, we calculate the current algebra which, as the central extension of the {G(k)} algebra, should be regarded as the dynamical symmetry algebra for the system and is therefore more important than the {G(k)} algebra. Finally, we demonstrate the existence of a kind of converse of Noether's theorem.

Locally hamiltonian systems with symmetry and a generalized Noether's theorem

An analysis of global aspects of the theory of symmetry groupsG of locally Hamiltonian dynamical systems is carried out for particular cases either of the symmetry group, or the differentiable manifoldM supporting the symplectic structure, or the action ofG onM. In every case we obtain a generalization of Noether's theorem. We also look at the classical Noether's theorem for Lagrangian systems from a modern perspective.

THE TRANSFORMATION PROPERTIES OF CONSTRAINED SYSTEM

The motion of constrained system may be described in terms of non-independent state functions. We considered the change of the action integral and constraint equations under the infinitesimal transformation of the time-space points and state functions of the constrained system, this leads to the general transformation result. We obtained the general equations for the transformation properties of constrained system along the trajectory of motion of that system. From these equations, the condition under which the transformation of constrained system yields conservation law can be worked out. For some specific transformations of the continuous system, we have found the equations of transformation properties. In the special cases, the general equations of transformation properties can be reduced to the results the classical Noether's theorem, The application of general results to classical mechanics is discussed in detail and the Poincare-Cartan invariant is generalized to the constrained system.