Noether's Theorem and the Conservation Laws of the KORTEWEG-DE VRIES Equation

A method developed recently by the author to derive a continuum of conservation laws by Noether's theorem from the so-called extended Bäcklund transformations is applied to the KORTEWEG -DE VRIES equation that describes various nonlinear dispersive wave phenomena in hydrodynamics, plasma physics and solid state physics. Further applications of Noether's theorem concerning this equation are given. It is shown that the Galilean transformation in the present case has an analogous function as Lie's transformation has with respect to the sine-Gordon equation.     

Revisiting Noether's Theorem on constants of motion

In this paper we revisit Noether's theorem on the constants of motion for Lagrangian mechanical systems in the ODE case, with some new perspectives on both the theoretical and the applied side. We make full use of invariance up to a divergence, or, as we call it here, Bessel-Hagen (BH) invariance. By recognizing that the Bessel-Hagen (BH) function need not be a total time derivative, we can easily deduce nonlocal constants of motion. We prove that we can always trivialize either the time change or the BH-function, so that, in particular, BH-invariance turns out not to be more general than Noether's original invariance. We also propose a version of time change that simplifies some key formulas. Applications include Lane-Emden equation, dissipative systems, homogeneous potentials and superintegrable systems. Most notably, we give two derivations of the Laplace-Runge-Lenz vector for Kepler's problem that require space and time change only, without BH invariance, one with and one without use of the Lagrange equation.     

Dynamical versus variational symmetries: understanding Noether's first theorem

It is argued that awareness of the distinction between dynamical and variational symmetries is crucial to understanding the significance of Noether's 1918 work. Special attention is paid, by way of a number of striking examples, to Noether's first theorem, which establishes a correlation between dynamical symmetries and conservation principles.     

Ableitung einer kontinuierlichen Mannigfaltigkeit von Erhaltungssätzen der modifizierten Korteweg-de-Vries-Gleichung nach dem Noetherschen Satz

Derivation of a Continuous Set of Conservation Laws for the Modified Korteweg-de Vries Equation by Noether's Theorem A method developed recently to derive a continuous set of conservation laws from extended Bäcklund transformations by means of Noether's theorem is applied to the modified Korteweg-de Vries (m. KdV) equation that describes Alfvén waves in a plasma. The corresponding conserved currents are equivalent to those found by WADATI , SANUKI and KONNO . It is shown that the extended Bäcklund transformation B?_α for the m. KdV equation, which coincides with that for the sine-Gordon equation, by MIURA'S transformation becomes the extended Bäcklund transformation β_x for the Korteweg-de Vries equation where x = 1/2α.     

The variational symmetries and conservation laws in classical theory of Heisenberg (anti)ferromagnet

The non-linear partial differential equations describing the spin dynamics of Heisenberg ferro- and antiferromagnet are studied by Lie transformation group method. The generators of the admitted variational Lie symmetry groups are derived and conservation laws for the conserved currents are found via Noether's theorem.     

Groups of canonical transformations and the virial-Noether theorem

In Hamiltonian mechanics a characterization of the infinitesimal generator of one-parameter Lie Groups of non-univalent canonical transformations is given. The result is used to derive a general form of the virial theorem, which has Noether's theorem as a special case. The theory is applied to the Toda lattice system.     

On symmetries,reductions, conservation laws and conserved quantities of optical solitons with inter-modal dispersion

This paper studies the compressional dispersive Alfvén (CDA) waves where Noether symmetries will be calculated from which the corresponding conservation laws will be obtained via Noether's theorem. Furthermore, one case of double reduction is performed via the association of a conserved vector with a Noether symmetry (with zero gauge). The conserved quantities of optical solitons in the presence of intermodal dispersion that is governed by the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity. The invariance-multiplier method is adopted to carry out the analysis, from which the conserved densities are then retrieved. Finally, the conserved quantities are obtained using the 1-soliton solution of the governing equation.     

Approximate first integrals of nonlinear oscillators with one-degree of freedom

Exact symmetries of the unperturbed (linear) part of the dynamical systems are determined. Resonance conditions which lead to the symmetry-breaking of the symmetries of the unperturbed part are obtained. The second-order approximate symmetries of the one degree of freedom, damped-driven oscillators are found. By employing an approximate version of Noether's theorem, second-order approximate first integrals are obtained for undamped oscillators. The results are discussed on the contour plots of the first integrals.     

Direct construction of a cubic selfinteraction for higher spin gauge fields

Using Noether's procedure we directly construct a complete cubic selfinteraction for the case of spin s=4$s=4$ in a flat background and discuss the cubic selfinteraction for general spin s with s derivatives in the same background. The leading term of the latter interaction together with the leading gauge transformation of first field order are presented.     

General trilinear interaction for arbitrary even higher spin gauge fields

Using Noether's procedure we present a complete solution for the trilinear interactions of arbitrary spins s₁$s_1$, s₂$s_2$, s₃$s_3$ in a flat background, and discuss the possibility to enlarge this construction to higher order interactions in the gauge field. Some classification theorems of the cubic (self)interaction with different numbers of derivatives and depending on relations between the spins are presented. Finally the expansion of a general spin s gauge transformation into powers of the field and the related closure of the gauge algebra in the general case are discussed.     

Newtons „Principia Mathematica Philosophia”︁ und Plancks Elementarkonstanten

Newton's “Principia Mathematica Philosophia” and Planck's Elementary Constants Together with Planck's elementary constants Newton's principles prove a guaranteed basis of physics and “exact” sciences of all directions. The conceptions in physics are competent at all physical problems as well as technology too. Classical physics was founded in such a way to reach far beyond the physics of macroscopic bodies.     

CONSERVATION LAWS FOR THE SCHRÖDINGER–NEWTON EQUATIONS

In this Letter a first-order Lagrangian for the Schrödinger–Newton equations is derived by modifying a second-order Lagrangian proposed by Christian [Exactly soluble sector of quantum gravity, Phys. Rev. D 56(8) (1997) 4844–4877]. Then Noether's theorem is applied to the Lie point symmetries determined by Robertshaw and Tod [Lie point symmetries and an approximate solution for the Schrödinger–Newton equations, Nonlinearity 19(7) (2006) 1507–1514] in order to find conservation laws of the Schrödinger–Newton equations.     

Newtons Wissenschaftslehre als Basis der Quantenphysik

Newton's Epistemology as Basic Concept of Quantum Physics It is correct to say that quantum physics cannot be derived from classical physics, which is founded on Newton's principles. However, it is also correct that Newton's epistemology, a more developed Platonian one, can be considered as basic for quantum physics. That is previously shown. Here, we remember Newton's epistemology more thoroughly, and consider particularly the difference to the Cartesian epistemology, a difference often veiled in the Newton tradition. Finally, we apply the result on some phenomena of quantum optics.     

The tetralogy of Birkhoff theorems

We classify the existent Birkhoff-type theorems into four classes: first, in field theory, the theorem states the absence of helicity 0- and spin 0-parts of the gravitational field. Second, in relativistic astrophysics, it is the statement that the gravitational far-field of a spherically symmetric star carries, apart from its mass, no information about the star; therefore, a radially oscillating star has a static gravitational far-field. Third, in mathematical physics, Birkhoff’s theorem reads: up to singular exceptions of measure zero, the spherically symmetric solutions of Einstein’s vacuum field equation with $\Lambda =0$ can be expressed by the Schwarzschild metric; for $\Lambda \ne 0$ , it is the Schwarzschild-de Sitter metric instead. Fourth, in differential geometry, any statement of the type: every member of a family of pseudo-Riemannian space-times has more isometries than expected from the original metric ansatz, carries the name Birkhoff-type theorem. Within the fourth of these classes we present some new results with further values of dimension and signature of the related spaces; including them are some counterexamples: families of space-times where no Birkhoff-type theorem is valid. These counterexamples further confirm the conjecture, that the Birkhoff-type theorems have their origin in the property, that the two eigenvalues of the Ricci tensor of 2-D pseudo-Riemannian spaces always coincide, a property not having an analogy in higher dimensions. Hence, Birkhoff-type theorems exist only for those physical situations which are reducible to 2-D.     

Geometry of Lagrangian First-order Classical Field Theories

We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in consideration. First of all, we construct all the geometric structures associated with a first-order jet bundle and, using them, we develop the lagrangian formalism, defining the canonical forms associated with a lagrangian density and the density of lagrangian energy, obtaining the Euler-Lagrange equations in two equivalent ways: as the result of a variational problem and developing the jet field formalism (which is a formulation more similar to the case of mechanical systems). A statement and proof of Noether's theorem is also given, using the latter formalism.     

Fractional vector calculus and fractional Maxwell’s equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’s theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered.