Showing papers in "Archive for Rational Mechanics and Analysis in 1968"

A general theory of heat conduction with finite wave speeds

Abstract: > 0 is a constant. This equation, which is parabolic, has a very unpleasant feature: a thermal disturbance at any point in the body is felt instantly at every other point; or in terms more suggestive than precise, the speed of propagation of disturbances is infinite. In this paper we develop a general theory of heat conduction for nonlinear materials with memory, a theory which has associated with it finite propagation speeds. In Section 3 we determine the restrictions that thermodynamics places on our constitutive relations. We show that our theory differs f rom other theories of heat conduction in that the heat-flux, like the entropy, is determined by the functional for the free-energy. In Section 6 we study the propagation of certain types of weak discontinuities. We show that in certain circumstances waves travelling in the direction of the heat-flux vector propagate faster than waves travelling in the opposite direction. In Section 7 we deduce the linearized theory appropriate to infinitesimal temperature gradients. We show that the linearized constitutive equation for the heat-flux q has the form: 1

Some theorems in classical elastodynamics

Abstract: This investigation is concerned with various fundamental aspects of the linearized dynamical theory for mechanically homogeneous and isotropic elastic solids. First, the uniqueness and reciprocal theorems of dynamic elasticity are extended to unbounded domains with the aid of a generalized energy identity and a lemma on the prolonged quiescence of the far field, which are established for this purpose. Next, the basic singular solutions of elastodynamics are studied and used to generate systematically Love's integral identity for the displacement field, as well as an associated identity for the field of stress. These results, in conjunction with suitably defined Green's functions, are applied to the construction of integral representations for the solution of the first and second boundary-initial value problem. Finally, a uniqueness theorem for dynamic concentrated-load problems is obtained.

Prolegomena to the rational analysis of systems of chemical reactions II. Some addenda

Abstract: Since the publication of some prolegomena to the rational analysis of systems of chemical reaction has evolved other cognate work has come to light and some earlier statements have been made more precise and comprehensive. This chapter is divided into three sections: (1) it discusses some publications that partially fill some of the undeveloped areas noticed before, (2) it extends the theorem on the uniqueness of equilibrium to a more general case and to establish the conditions for the consistency of the kinetic and equilibrium expressions,, and (3) the conception of a reaction mechanism is to be reformulated in a more general way and the metrical connection between the kinetics of the mechanism and those of the ostensible reactions clarified. The notation of the earlier paper is followed and augmented where necessary—in particular, the range of each affix is carefully specified and the summation convention of tensor analysis is employed.

Asymptotic Formulas with Remainder Estimates for Eingevalues of Elliptic Operators

Abstract: We propose to discuss in this lecture a number of results related to the problem of eigenvalue distribution of elliptic operators. We start with some classical results. Let Δ be the Laplacian in Rn and consider the eigenvalue problem: $$\begin{array}{*{20}c}{ - \Delta = \lambda {\text{u}}} & {{\text{in}}\,\,\,\Omega \,,}\\{{\text{u}} = 0} & {{\text{on}}\,\,\partial \Omega \,,}\\\end{array}$$ (1)

Singularities of the n-body problem. I

Abstract: Little is known about the nature of the singularities of the n-body problem. While it is plausible to suppose that they are due to collisions, this has never been established, except when n = 2 or n = 3. In the general case the best that can be said at present is the fact, due to PAINLEV~ [5], that a singularity occurs at the time to if and only if the minimum of the mutual distances between pairs of particles approaches zero as the time t approaches to. In the present paper we shall investigate the problem of singularities due to collisions. We define a singularity at time to to be due to collisions i f as t ~ to each particle approaches a definite position in the inertial coordinate f rame. This means, in view of PAINLEV~'S theorem, that at least two particles approach the same point. In 1908 VON ZEIPEL [4] published a statement to the effect that if the system remains bounded as t--+ to, then a singularity at time t o is due to collisions. His proof is erroneous, and the assertion still stands as a conjecture. The purpose of the present paper is to obtain necessary and sufficient conditions for a singularity due to collisions. It will be supposed that the origin of coordinates is fixed at the center of mass, and that the singularity occurs as t ~ 0 +. The following notation will be used. The symbols m k, re, vk denote respectively the mass, position and velocity of the kth particle. We define further