Energy Conditions and Conservation Laws in LRS Bianchi Type Ⅰ Spacetimes via Noether Symmetries

In this paper, we have completely classified the locally rotationally symmetric(LRS) Bianchi Type Ⅰ spacetimes via Noether symmetries(NS). The usual Lagrangian corresponding to LRS Bianchi Type Ⅰ metric is used to find the set of determining equations. To achieve a complete classification, these determining equations are generally integrated to find the components of NS vector field and the metric coefficients. During this procedure, several cases arise which give different Noether algebras of dimension 5,..., 9, 11, and 17. A comparison is established between the obtained NS and the Killing and homothetic vectors. Corresponding to all NS generators, the conservation laws are stated by using Noether's theorem. The metrics which we have obtained as a result of our classification are shown to be anisotropic or perfect fluids which satisfy certain energy conditions.     

Stern-Gerlach Experiment’s Interpretations and Noether’s Theorem

In this paper we suggest that theories treating two interacting objects in a different manner (for instance electromagnetic field of a laser classically, and the interacting atom as a quantum object) should be called “mixed”. Mixed theories are not so rare in Physics. One just should look at the whole area of Atomic, Molecular and Optical Physics in which mixed theories are often used, and, also, theories including quantum object interacting with classical surroundings that are the subject of our present discussion: the field of Quantum decoherence, when applied to resolving the dilemma should classical trajectories be used in explaining the Stern-Gerlach experiment or not. Consequently we are proving one improved corollary to Noether’s theorem, stating that mixed theories are not supporting the law of conservation of angular momentum and spin, as they are not based on the isotropy of space-time.     

Conservation Laws for the Derivative Nonlinear Schroedinger Equation with Non-vanishing Boundary Conditions

Conservation laws for the derivative nonlinear Schr6dinger equation with non-vanishing boundary conditions are derived, based on the recently developed inverse scattering transform using the affine parameter technique.     

Gauge Transformation and Conservation Laws of Landau-Lifschitz Equation for a Spin Chain

Introducing an explicit expression of gauge transformation turning the spin in linear order of spectral parameter in the first Lax pair into the third axis in spin space, we derive the required Hamiltonian densities for the spin chain with the easy axis, easy plane, and full anisotropy, respectively.     

Lie Symmetries,Conservation Laws and Explicit Solutions for Time Fractional Rosenau–Haynam Equation

Under investigation in this paper is the invariance properties of the time fractional Rosenau-Haynam equation, which can be used to describe the formation of patterns in liquid drops. By using the Lie group analysis method, the vector fields and symmetry reductions of the equation are derived, respectively. Moreover, based on the power series theory, a kind of explicit power series solutions for the equation are well constructed with a detailed derivation. Finally, by using the new conservation theorem, two kinds of conservation laws of the equation are well constructed with a detailed derivation.     

Conservation Laws for Partially Conservative Variable Mass Systems via d＇Alembert＇s Principle

Conservation laws for partially conservative variable mass dynamical systems under symmetric infinitesimal transformations are determined. A generalization of Lagrange-d ＇Alembert＇s principle for a variable mass system in terms of asynchronous virtual variation is presented. The generalized Killing equations are obtained such that their solution yields the transformations and the associated conservation laws. An example illusstrative of the theory is furnished at the end as well.     

Double Reduction and Exact Solutions of Zakharov–Kuznetsov Modified Equal width Equation with Power Law Nonlinearity via Conservation Laws

The conservation laws for the （1＋2）-dimensional Zakharov-Kuznetsov modified equal width （ZK-MEW） equation with power law nonlinearity are constructed by using Noether＇s approach through an interesting method of increasing the order of this equation. With the aid of an obtained conservation law, the generalized double reduction theorem is applied to this equation. It can be shown that the reduced equation is a second order nonlinear ODE. FinaJ1y, some exact solutions for a particular case of this equation are obtained after solving the reduced equation.     

Weyl’s Law and Connes’ Trace Theorem for Noncommutative Two Tori

We prove the analogue of Weyl’s law for a noncommutative Riemannian manifold, namely the noncommutative two torus ${\mathbb{T}_{\theta}^{2}}$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on ${\mathbb{T}_{\theta}^{2}}$ . We also prove the analogue of Connes’ trace theorem by showing that the Dixmier trace and a noncommutative residue coincide on pseudodifferential operators of order ?2 on ${\mathbb{T}_{\theta}^{2}}$ .     

Field Equations and Conservation Laws Derived from the Generalized Einstein’s Lagrangian Density for a Gravitational System and Their Influences upon Cosmology

Through discussions on the fundamental properties of the generalized Einstein’s Lagrangian density for a gravitational system, the theoretical foundations of the modified Einstein’s field equations and the Lorentz and Levi-Civita’s conservation laws are systematically studied. The theory of cosmology founded on them is discussed in detail and some new properties and new effects of the cosmos are deduced; these new properties and new effects could be tested via future experiments and observations.     

Noether’s Theorem and its Inverse of Birkhoffian System in Event Space Based on Herglotz Variational Problem

International Journal of Theoretical Physics - Herglotz variational principle, in which the functional is defined by a differential equation, generalizes the classical ones defining the functional...     

Exactly Solvable Combinations of Scalar and Vector Potentials for the Dirac Equation Interrelated by Riccati Equations

New classes of solvable scalar and vector potentials for the Dirac equation are obtained, together with the associated exact Dirac spinors. The method of derivation is based on an a priori constraint between the solutions, leading to an interrelation between the scalar and vector potential in the form ofa Riccati equation. The present note generalizes a series of former articles.     

Jacobi’s Last Multiplier and the Complete Symmetry Group of the Ermakov-Pinney Equation

Abstract

The Ermakov-Pinney equation possesses three Lie point symmetries with the algebra sl(2, R). This algebra does not provide a representation of the complete symmetry group of the Ermakov-Pinney equation. We show how the representation of the group can be obtained with the use of the method described in Nucci, J. Nonlin. Math. Phys. 12 (2005) (this issue), which is based on the properties of Jacobi’s last multiplier (Bianchi L, Lezioni sulla teoria dei gruppi continui finiti di trasformazioni, Enrico Spoerri, Pisa, 1918), the method of reduction of order (Nucci,J. Math. Phys 37 (1996), 1772–1775) and an interactive code for calculating symmetries (Nucci, Interactive REDUCE programs for calcuating classical, non-classical and Lie-Bäcklund symmetries for differential equations (preprint: Georgia Institute of Technology, Math 062090-051, 1990, and CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 3: New Trends in Theoretical Developments and Computational Methods, Editor: Ibragimov N H, CRC Press, Boca Raton, 1996, 415–481).