Showing papers on "Conservation law published in 1968"

Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion

Abstract: With extensive use of the nonlinear transformations presented in Paper I of the series, a variety of conservation laws and constants of motion are derived for the Korteweg‐de Vries and related equations. A striking connection with the Sturm‐Liouville eigenvalue problem is exploited.

New Conservation Laws for Zero Rest-Mass Fields in Asymptotically Flat Space-Time

Abstract: Some recently discovered exact conservation laws for asymptotically flat gravitational fields are discussed in detail. The analogous conservation laws for zero rest-mass fields of arbitrary spin s(= 0, $\frac{1}{2}$, 1, $\ldots$) in flat or asymptotically flat space-time are also considered and their connexion with a generalization of Kirchoff's integral is pointed out. In flat space-time, an infinite hierarchy of such conservation laws exists for each spin value, but these have a somewhat trivial interpretation, describing the asymptotic incoming field (in fact giving the coefficients of a power series expansion of the incoming field). The Maxwell and linearized Einstein theories are analysed here particularly. In asymptotically flat space-time, only the first set of quantities of the hierarchy remain absolutely conserved. These are 4s + 2 real quantities, for spin s, giving a D(s, 0) representation of the Bondi-Metzner-Sachs group. But even for these quantities the simple interpretation in terms of incoming waves no longer holds good: it emerges from a study of the stationary gravitational fields that a contribution to the quantities involving the gravitational multipole structure of the field must also be present. Only the vacuum Einstein theory is analysed in this connexion here, the corresponding discussions of the Einstein-Maxwell theory (by Exton and the authors) and the Einstein-Maxwell-neutrino theory (by Exton) being given elsewhere. (A discussion of fields of higher spin in curved space-time along these lines would encounter the familiar difficulties first pointed out by Buchdahl.) One consequence of the discussion given here is that a stationary asymptotically flat gravitational field cannot become radiative and then stationary again after a finite time, except possibly if a certain (origin independent) quadratic combination of multipole moments returns to its original value. This indicates the existence of 'tails' to the outgoing waves (or back-scattered field), which destroys the stationary nature of the final field.

Related First Integral Theorem: A Method for Obtaining Conservation Laws of Dynamical Systems with Geodesic Trajectories in Riemannian Spaces Admitting Symmetries

Abstract: In this paper we develop in detail a unified method, referred to as the Related First Integral Theorem, for obtaining ``derived'' first integrals (i.e., constants of the motion) of mass‐pole test particles with geodesic trajectories in a Riemannian spece. By this method, which is based upon a process of Lie differentiation, additional conservation laws in the form of mth order first integrals can be generated from a given mth order first integral (conservation law), provided the space admits symmetries in the form of continuous groups of projective collineations (which include affine collineations and motions as special cases). We give in tensor form a reformulation of the well‐known Poisson's theorem on constants of the motion for particles with geodesic trajectories. We then show for this class of trajectories that, as a method for generating mth order first integrals from a given mth order first integral, Poisson's theorem is a special case of the Related First Integral Theorem. It is also shown that d...

Some general properties of para-fermi field theory.

Abstract: The nonrelativistic theory of a single para-Fermi field of order $p$ is investigated. General properties of state vectors are studied in detail, and it is shown that the state-vector space can be spanned by what we shall call standard state vectors. A restriction on the form of interaction Hamiltonians is derived from the requirement that our formalism be described by local Lagrangian field theory. This restriction on interaction Hamiltonians gives rise to a conservation law of a physical quantity to be called $A$, which resembles the magnitude of angular momentum with respect to its rule of addition. The conservation law of $A$ leads then to absolute selection rules for reactions, which are a generalization of those obtained elsewhere. The problem of bound states made up of our para-Fermi field is also studied, and all bound states are classified into ($p+1$) categories according to their statistical behaviors. It is found that for $pl~3$ all bound states can be described by ordinary parafield theory, whereas for $pg~4$ such is no longer the case. Furthermore, it can be shown that in the theory of $p=2$ no fermion bound states are possible. In this sense it may be said that para-Fermi fields of $p=1$ and 3 occupy a very privileged position in para-Fermi theory in general. The main results in this paper are stated as 12 theorems. It is expected that the whole argument will be valid in a relativistic theory as well.

Time-Translation Invariance for Certain Dissipative Classical Systems

Abstract: Time-translation invariance of the equation of motion for an arbitrary one-dimensional classical system has been shown to generate a conservation law. For conservative systems, this conserved quantity is shown here to be the usual energy. For the linearly damped harmonic oscillator, the explicit form of this constant is found, as well as for quadratically damped systems. In both cases, as the damping approaches zero, the conserved quantity is shown to approach a function of the usual energy. For both types of dissipation, Lagrangians are given which yield the equations of motion.

Green's Theorem and Invariant Transformations

Abstract: Conservation laws are derived with the use of Green's theorem. These are studied specifically for the wave equation. A trivial class of invariant transformations exists which maps solutions into solutions. The mapping consists in the addition of a particular solution to all solutions. Because of the linearity of the wave equation, this sum will again be a solution. It is shown that this class of transformations has the Newman‐Penrose constants among its generators. Calculations are carried out explicitly for the scalar wave equation and for Maxwell's equations.

Momentum of Longitudinal Elastic Vibrations

Abstract: The field-theoretic momentum of longitudinal vibrations in an elastic medium is not the same as the total momentum of the particles which constitute the medium. In this article, we discuss the connection between these two quantities in terms of a new conservation law which follows from an underlying symmetry of the system. The surprising feature of this conservation law is that it is expressed in terms of variables which refer to the medium only, and yet remains valid under a very wide class of couplings of the medium to external systems. The analysis is carried out within the context of a simple one-dimensional model which is easily generalized to three dimensions.

Relativistic invariance without angular-momentum conservation

Abstract: We show that, contrary to a rather widespread belief, relativistic invariance does not necessarily lead to angular-momentum conservation. Noticing that there is no empirical evidence for such a conservation in weak interactions, we show that a new classification of strange particles is possible in which the K-meson has spin 1/2, the Λ and ∑ have spin 1 and the Ξ has spin 3/2. Our scheme can be proved or disproved with simple experiments.

Global solutions of hyperbolic systems of conservation laws in two dependent variables

Abstract: The vector U= (v, u) is a function of / and x> t^O, — oo < # < oo, and the functions/and g are C functions of two real variables. We assume that the system (1) is hyperbolic in some open set 01 in the v-u plane, with fvgu>0. Let DF(U) and D F(U) denote respectively the first and second Fréchet derivatives (see [2]) of the vector function F= (f, g) : Ol-»i?; and let rj(U)J = li 2, be the eigenvectors olDF(U), with orthogonal vectors lj(U)1 j= 1, 2: k(U)rj(U) = 0 for i^j.

Isospin and number-conserving treatment of pairing correlations in nuclei

Abstract: A previous study of the pairing interaction between identical particles by means of the equations of motion is extended to the case of unlike nucleons interacting through a charge-independent force. By comparison with the Hecht model for which exact numerical results are available, our results are seen to be a compromise between those of the B.C.S. and Lipkin-Nogami methods.

A Theoretical Approach to Two-Phase Critical Flow : (3rd Report, The Critical Condition Including Interphasic Slip)

Abstract: "Eigenvalue method", already proposed by the author for two-phase critical flow, is developed for the axial steady flow equations system where separate momentum conservation can be strictly postulated. Besides the comparisons of physical meanings in two-phase critical flow between energy and entropy, conservation equations are also investigated. The conclusions for steam water mixtures are as follows. (1) "Eigenvalue method"defines the criticality as the discontinuous condition of steady flow differential equations system and can treat easily such a complicated system composed of separate momentum between two phases, energy, and mass conservation equations. (2) The solutions given by the above theory can well explain the former empirical data except for critical slip ratio which has not well been investigated experimentally. (3) Critical condition with energy conservation gives a more accurate solution than one with entropy conservation. But the latter is still good approximation.

Further consideration of energy and momentum in general relativity

Abstract: The problem of the proper formulation of the conservation laws of energy and linear momentum in general relativity is discussed The Komar expression, taken as a generally covariant conservation law generator, is considered in light of the problem which Moller found with his energy-momentum complex The conservation laws as formulated here from the Komar generator are shown to be devoid of such difficulties Also, a generally covariant formulation of the conservation laws, which are differentiated from a covariant conservation-law generator, is effectively achieved without the necessity of introducing a tetrad structure In this formulation the conservation lawsper se are noncovariant quantities; however, one may always return to the Komar generator and transform in a generally covariant manner any given conservation law, through its underlying symmetry representation, to any space-time co-ordinate system of interest It is also possible, at least in some simple space-time systems, to formulate the energy and momentum when they are not rigorously conserved entities Such expressions can have possible application, for example, in radiating gravitational systems

Symmetry and conservation laws in classical mechanics

Abstract: Ideas of symmetry form an essential part of modern physics. It seems advantageous to introduce these ideas to students at a relatively early stage by using them in the context of classical mechanics. The aim is to demonstrate the results that invariance under time and space translations and space rotations imply conservation of energy, momentum and angular momentum. But how does one teach this and just what is it that has to be invariant?