Showing papers on "Conservation law published in 1970"

Interaction between hydromagnetic waves and a time-dependent, inhomogeneous medium.

Abstract: Consider a system made up of a hydromagnetic wave and the slowly varying background fluid in which it propagates. It is shown that both the effect of the background on the wave and that of the wave on the background may be derived from Hamilton's principle using the averaged hydromagnetic Lagrangian density. The waves propagate adiabatically, conserving the wave action, and act on the background via a wave pressure term. Total momentum, angular momentum, and energy are conserved. When many waves are superimposed, as in weak turbulence, the wave kinetic equation replaces the adiabatic conservation equation. The accuracy of the averaging approximation is examined, and it is shown that it may be extended to all orders in the inhomogeneity. Also, Eulerian and Lagrangian averaging are discussed.

Korteweg‐deVries Equation and Generalizations. V. Uniqueness and Nonexistence of Polynomial Conservation Laws

Abstract: The conservation laws derived in an earlier paper for the Korteweg‐deVries equation are proved to be the only ones of polynomial form. An algebraic operator formalism is developed to obtain explicit formulas for them.

Theory of Electrodynamics in Media in Noninertial Frames and Applications

Abstract: The theory of local electrodynamics of media in given noninertial frames, within the Maxwell‐Einstein theory, is constructed in terms of local observable EM fields and physical media parameters of its comoving frame. Localization of tensors to observables for and their relations among observers in different frames are introduced; local and global constitutive tensors and local Maxwell equations are obtained and interpreted. Also, a Lagrangian formulation for both lossless and lossy media is constructed, and boundary conditions, local conservation laws, and energy tensors are obtained. The applications concern linear accelerational and rotational media in flat space‐time for which local Maxwell equations in comoving frames are obtained. Then an EM wave propagating in the direction of acceleration is studied in the accelerating frame. The first‐order propagation shows a frequency shift and amplitude change which have very simple physical significances of instantaneous Doppler shift and photon density in med...

Boltzmann Equation and Angular Momentum Conservation

Abstract: The form of the Boltzmann equation presently being used to describe phenomena dependent upon the internal angular momentum states of molecules in the gas phase is inconsistent in that angular momentum is not conserved. Thus, the internal angular momentum relaxation is correctly described by this equation but the resulting production of angular momentum in the translational degrees of freedom just does not appear at all. This work is aimed at extending the Boltzmann equation to give a consistent description of all conserved quantities. It is shown that this is not a trivial matter and that some truncation of an expansion in position gradients is required. The simplest choice is discussed. In the development, a central role is played by sum rules which arise from the assumed localized nature of the intermolecular potential.

Law of Conservation of the Capital-Output Ratio

Abstract: Just as simple harmonic motion, definable by a variational condition, δ [unk] (½ ẋ2 - ½ x2) dt = 0, has motions which must conserve the sum of kinetic and potential energies, ½ ẋ2 + ½x2 ≅ constant, so in a neoclassical von Neumann economy, where all output is saved to provide capital formation for the system's growth, it will be true that there exists a conservation law—namely the constancy along any intertemporally-efficient motion of the capital-output ratio ΣPtjKtj/ΣPtjKt j. This is derived as an „energy” integral of a time-free integrand1 in an optimal-control problem of variational type.

Conservation laws in general relativity based upon the existence of preferred collineations

Abstract: The conservation laws based upon the existence of curvature and Ricci collineations are investigated and the results given recently by Katzin, Levine and Davis are reinterpreted and generalized. The concept of a ‘Maxwell collineation’ is introduced and corresponding conservation laws are found.

Self-consistent approximations for itinerant ferromagnets above the phase transition point

Abstract: The “conserving and self-consistent” approximation scheme of Kadanoff and Baym is generalized to systems of particles of non-zero spin. The additional conservation law of the total spin of the system restricts the possible approximations for the self-energy part, and thus the approximations for the irreducible vertex part occuring in the equation for the correlation function. This guarantees, for instance, the correct behavior of the dynamical susceptibility in the long wave length limit. Two examples are discussed under these aspects: the Hartree-Fock and theT-matrix approximation for the self-energy part and the resulting susceptibility.

Longitudinal motion of an elastic bar

Abstract: The exact equations of motion of an elastic bar are discussed, both in material and local coordinates. It is shown that for an ideal elastic material the former, but not the latter, are linear. An infinite number of conservation laws is shown to exist.

Discrete spectra and dampes waves in quasilinear theory

Abstract: Quasi-linear equations consequences in discrete spectra and damped electron plasma waves, discussing conservation laws relation to resonance approximation

Selfconsistent approximations for the Heisenberg model

Abstract: The Baym-Kadanoff method for generating conserving approximations is generalized so that it may be applied to the drone-fermion representation of the spin 1/2 Heisenberg model. It is shown that the generalized procedure gives approximations in which the correlation functions preserve (i) commutation rules between spin operator components, (ii) conservation laws for the total spin, (iii) the relation between long wavelength longitudinal susceptibility and the field derivative of the magnetization, and (iv) the isotropy of the system in the paramagnetic phase in zero field. Moreover the approximate free energy is stationary with respect to variations of the Green function, and its derivatives with respect to external parameters are correctly related to the correlation functions.

Four-Point Method of Characteristics

Abstract: Conservation laws in two independent variables have a four-point symmetry when expressed in normal or characteristic form. This symmetry is notated for nearly horizontal flows and translated into easily programmed numerical operators. Simple starting and boundary subroutines are described. The output is especially well adapted to digital-analog plotting techniques, with an accuracy superior to any current difference scheme. Bed slopes and resistance terms can be easily introduced, but the method becomes less viable for general stream flows, with changes of channel section.

Charge moving with the speed of light.

Abstract: I HAVE recently shown1 that Maxwell's equations admit solutions for charge moving in a straight line with the speed of light c. The accompanying electromagnetic field Fik is that of a plane-fronted wave, that is, one whose front is plane and whose amplitude varies over the front. The source is a null four-current, Ji satisfying where gik is the metric tensor of flat space-time. The charge is localized in a world-tube about the z-axis of coordinates; all appropriate conditions are satisfied at the boundary of the tube, and the Fik are everywhere continuous. Although charge is radiated away from the source, conservation of charge is of course automatically ensured by Maxwell's equations. The solutions have precise analogues in Einstein–Maxwell theory2.

The propagation law of cleavage fracture

Abstract: The equations of motion of cleavage fracture are solved and discussed for constant force, constant moment, and constant deflection. Also crack propagation in a plate is discussed. The differential equations are derived from the law of angular momentum conservation of the split parts. An energy balance, which includes the kinetic energy of the moving crack is incorporated into the law of angular momentum conservation. The static case of crack stability is a particular solution of the general propagation equation.

Momentum conservation, time dependent Ginzburg Landau equation and paraconductivity

Abstract: Superconductors exhibit increasing electrical conductivity as the temperature approachesT c from above, due to superconducting fluctuations. The functions σf1=σ(ω, ɛ)-σ n (ω), ɛ=(T-T c )/T c , have been derived by Schmidt phenomenologically using the time dependent Ginzburg-Landau equation (TDGL). These functions fail to vanish in the absolute clean limit τ → ∞ as they must. We have therefore reinvestigated the derivation of the linearized TDGL-equation and the corresponding current expression in the presence of a time dependent vector potential. We find several new terms, which are important for the rather clean superconductor only and are easily interpreted physically in terms of momentum conservation. Applying these corrected equations to the paraconductivity problem, we derive σfl(ω, ɛ) which has an extra factor (1 —iωτ)−2 compared to Schmidt's result. There is also an additional term, which is connected to the problem of the contribution calculated by Maki. By comparison with the linear response function belowT c , we show that this term is valid in the limit ¦ω¦≫¦Δ¦ only and may not be continued to ω=0. There remains, however, a problem connected with this term, which cannot be solved within the present phenomenological framework.

Three-dimensional second-order accurate difference schemes for discontinuous hydrodynamic flows

Abstract: In this paper, we show how a second-order accurate two-step method used in the numerical computation of hydrodynamic flows may be derived directly from the integral conservation laws. A necessary and sufficient condition for stability for the linearized equations is derived for the three-dimensional Cartesian coordinate case. I. Introduction. In 1960 Lax and Wendroff (2) presented a second-order accurate scheme for the numerical computation of hydrodynamic flows (neglecting various stresses and heat conduction). An important feature of the method was that the differential equations were written as a first-order system in conservation form. The difference scheme was derived by expanding the solution in a Taylor series in the time variable up to terms of second order. The method involved the computation of matrices, the determinants of which were the Jacobians of certain transforma- tions. Richtmyer (3) presented a two-step method, for problems in two-space dimen- sions, explicit like the Lax-Wendroff, but which required no matrix calculations and had the same order of accuracy. A third explicit two-step method, avoiding matrix calculations, was used by Rubin and Burstein (4) and in the latter paper all three schemes were compared. In Section II of this paper, we shall show how the two- step Lax-Wendroff method as given by Richtmyer can be generalized to three- space dimensions and time. The result follows, with suitable approximations, directly from the integral conservation laws. The motivation for this approach is twofold: (1) The conservation laws are, in fact, integral in nature.** (2) Alternative quadrature methods immediately suggest themselves. In par- ticular, we have used the midpoint and rectangular rules for numerical integration. This approach, however, may be used to derive approximations of higher order than two. It is to be noted that the analysis given here depends on the differentia- bility of the functions. The applications of interest, however, are to discontinuous flows. The justification for the use of this approximation is given in (2) where it is proved that if a solution to the finite-difference equations exists, then the Lax- Wendroff scheme converges to a weak solution of the conservation laws. In (1), Anderson, Preiser, and Rubin showed how the hydrodynamic equations could be written in conservation form for arbitrary orthogonal curvilinear coordinate sys-

Conservation laws in the general theory of relativity

Abstract: Investigation of the relationship between conservation laws in general relativity and certain types of « generalized symmetry properties » (generated by transformations which map space-times onto other space-times which are not describable as co-ordinate mappings) yields, in contrast to the case for co-ordinate mappings, a very restricted class of possible formal conservation expressions. Here we consider in particular the affine, projective, and conformal correspondence of Riemannian space-times. Particle and field conservation-law generators are formulated as well as the conditions necessary for conservation laws and related symmetry properties to be admitted. The particle case is illustrated by a well-known quadratic first integral of the geodesic equations of the Friedmann-Lemaitre cosmological space-time which follows in consequence of a projective symmetry property (geodesic correspondence). The field case is illustrated by formulating a conservation law following in consequence of a conformal symmetry property manifested by the familiar plane gravitational wave solutions which are included by some of the earlier work of Brinkmann.

Electromagnetic-transition form factors free of kinematic singularities

Abstract: Matrix elements of the electromagnetic currents between one-particle states of spins ands′ are decomposed into invariant functions free of kinematic singularities which are the usual multipole amplitudes in the nonrelativistic limit. Parity conservation is taken into account. Kinematic constraints occur only at the two physical threshold values and are given explicitly.

Mechanism for the breakdown of adiabatic invariance.

Abstract: Computation of particle orbits in systems with more than one degree of freedom shows that adiabatic invariance is destroyed due to a resonance between the oscillation periods associated with two degrees of freedom. These resonances can be predicted qualitatively from first‐order perturbation theory, which also indicates that the breakdown is related to the precentage change in the invariant during one period of the motion.

The adiabatic electron plasma and its equation of state

Abstract: For slow processes in a collisionless two-component plasma which conserve the adiabatic invariant of the electrons, the equation of motion for electrons is derived. The conservation laws for the whole system are discussed in relation to the equation of state and a consistent distribution function for the electrons.

Post-Newtonian Methods and Conservation Laws

Abstract: The present paper is concerned with the conservation laws in general relativity as expressed in terms of the Landau-Lifshitz complex and their role in the development of the successive post-Newtonian approximations to the equations of general relativity of a perfect fluid. In § I, the known conservation laws of Newtonian hydrodynamics are formulated in a language that the relativistic laws appear as natural generalizations. In § II, the same laws are considered in the framework of general relativity. In particular the conserved energy is identified as the difference between the (0,0)-component of the Landau-Lifshitz complex and the conserved rest-mass energy (\(={{c}^{2}}p{{u}^{0}}\sqrt{-g}\)). In § III, the development of the first and the second post-Newtonian approximations to the equations of relativistic hydrodynamics are described and illustrated. And finally in § IV, the manner in which one can obtain the equations of the 2 1/2 — post-Newtonian approximation is described. In this approximation all terms inclusive of O(c−5) beyond the Newtonian are retained; it is in this approximation that terms representing the reaction of the fluid to the emission of gravitational radiation by the system first make their appearance. It is shown how the derived radiation-reaction terms of O(c−5) contribute to the dissipation of energy and angular momentum in agreement with the predictions of the linearized theory of gravitational radiation.

Methods for deriving conservation laws

Abstract: Systematic methods are used to find all possible conservation laws of a given type for certain systems of partial differential equations, including some from fluid mechanics. The necessary and sufficient conditions for a vector to be divergence-free are found in the form of a system of first order, linear, homogeneous partial differential equations, usually overdetermined. Incompressible, inviscid fluid flow is treated in the unsteady two-dimensional and steady three-dimensional cases. A theorem about the degrees of freedom of partial differential equations, needed for finding conservation laws, is proven. Derivatives of the dependent variables are then included in the divergence-free vectors. Conservation laws for Laplace's equation are found with the aid of complex variables, used also to treat the two-dimensional steady flow case when first derivatives are included in the vectors. Conservation laws, depending on an arbitrary number of derivatives, are found for a general first order quasi-linear equation in two independent variables, using two differential operators, which are associated with the derivatives with respect to the independent variables. Linear totally hyperbolic systems are then treated using an obvious generalization of the above operators.

Geometrical Derivation of the Conservation Laws

Abstract: Conservation laws which are customarily obtained by invoking the invariance of a Lagrangian density under the transformations of some intrinsic symmetry group may be given a completely geometrical treatment within the context of the theory by Yang and Mills. The formalism is extended from unitary transformations to general linear transformations and the concepts of parallel transfer, covariant differentiation, and intrinsic curvature tensor are discussed. Conservation laws follow from the assumed invariance of the Lagrangian under parallel transfer defined with a general affine connection which is a direct sum of intrinsic and space‐time affinities. Conservation of generalized charge is a consequence of the arbitrariness of that part of the affinities which operates in the intrinsic space, and conservation of energy and momentum is related to the arbitrariness of the part of the affinities which operates on the space‐time indices. Some comments on Palatini's derivation of Einstein's equation of general relativity are made.