Motion Rules Research Articles

Dynamics of translations and rotations in molecular crystals: Macroscopic and microscopic approaches

Starting from a Hamiltonian for an orientationally disordered crystal with bilinear coupling between translational and orientational coordinates, a nonequilibrium macroscopic free energy is calculated. It is used to obtain coupled dynamic equations for translations and rotations by methods of irreversible thermodynamics. The corresponding dynamic response functions are in full agreement with those derived using a microscopic Mori-type projection-operator method. The coefficients of the macroscopic free energy are related to the static susceptibilities obtained from the Hamiltonian in the framework of molecular-field theory. Dynamic sum rules for translational-rotational motion are given.

Effective stiffness theory for a laminated elastic-viscoplastic work-hardening composite

Effective stiffness theory is derived for modeling the dynamic behavior of a laminated composite material made of elastic-viscoplastic work-hardening constituents. The resulting theory represents the composite as a higher order continuum with microstructure, with the corresponding equations of motion and flow rules. The theory is illustrated and applied to obtain the dynamic response of a laminated slab subjected to time-dependent normal extended load in the direction of the layering.

RATE OF CONVERGENCE IN THE "CIRCLE FORMATION" PROBLEM

We consider transformations obtained from local homogeneous rules of motion of polygons in the plane for which regular polygons are stationary (the so-called circle formation problem). These transformations are studied for initial states that are close to regular polygons. A method of determination of the rate of convergence to regular polygons is presented; on this basis we obtain estimates of the highest rate of convergence. Figures: 1. Bibliography: 4 items.

STUDY OF CONVERGENCE IN A RECTIFICATION PROBLEM

The problem of rectification is considered of the plane motion of a polygonal line specified by its vertices. The motion is defined by an operator in -dimensional space whose infinite repetition must rectify the polygonal line. The rules of motion of the vertices are local and homogeneous for all the internal vertices of the polygonal line. The behavior of in the neighborhood of stationary points is studied, and global convergence to these points from certain initial states is proved for .Figures: 4. Bibliography: 2 items.

Regarding stopping rules for Brownian motion and random walks

Regarding stopping rules for Brownian motion and random walks

Two-Dimensional Motion of Idealized Grain Boundaries

To represent ideal grain boundary motion in two dimensions, a rule of motion of plane curves is considered whereby any given point of a curve moves toward its center of curvature with a speed that is proportional to the curvature. A general theorem is deduced concerning the change of area enclosed by such a curve. Three families of curves are found that obey the curvature rule of motion while undergoing the shape preserving transformations of uniform magnification, translation, and rotation respectively. Pieces of these curves represent the steady shapes of idealized grain boundaries under certain symmetrical conditions.

二次元的流れの中の二次的循環についての覚え書

Squire and Winter solved secondary flow problem by replacing the boundary layer with vorticity. They used curvilinear coordinates and is pretty complex. Hawthorne presented another method founded on the same idea using vector analysis after that, but his method is also inaccessible.The writer presents in this paper an easy method based on the rule of vortex motion, and has obtained the same results of 1st approximation as that of Squire, Winter and Hawthorne.

Forecasting the Motion of Tropical Cyclones*

Application of theories of Bjerknes and Holmboe to the motion of tropical cyclones is attempted. The equation for east-west motion is presented and the equation for north-south motion is derived. Rules for motion of tropical cyclones are deduced from these equations, which involve the wind distribution and the geometry of the cyclone. Statistical evidence corroborates use of equations and rules.

Second essay on a general method in dynamics

This essay is a sequel of the one which appeared in the last volume of the Philosophical Transactions, and which contained a general me­thod for reducing all the most important problems of dynamics to the study of one characteristic function, or one central or radical relation. It was there remarked that many eliminations required by this me­thod might be avoided by a general transformation, introducing the time explicitly into a part (S) of the whole characteristic function (V) ; and the first object of the present essay is to examine and develope the properties of this part (S), which the author designates by the term Principal Function . This function is applied by the author to problems of perturbation, in which he finds it dispenses with many laborious and circuitous processes, and furnishes accurate expressions of the disturbed configurations of a system by the rules of undisturbed motion, if only the initial components of velocities be changed in a suitable manner. Another manner of extending rigorously to dis­turbed, the rules of undisturbed motion, by the gradual variation of elements, in number double the number of the coordinates or other marks of position of the system, which was first invented by Lagrange, and was afterwards improved by Poisson, is considered in this second essay under a form rather more general; and the general method of calculation which has already been applied by the author to other analogous questions in optics and in dynamics, is now applied to the integration of the equations which determine these elements. This general method is founded chiefly on a combination of the principle of variations with those of partial differentials, and may furnish, when matured, a separate branch of analysis, which may be denominated the Calculus of Principal Functions . When applied to the integra­tion of the equations of varying elements, it suggests the consideration of a certain Function of Elements , capable of being variously trans­formed, and which may be either rigorously determined, or at least approached to, by a corollary of the general method. With a view to illustrate these new principles, and more especially those connected with problems of perturbation, they are applied, in this essay, first, to a very simple example, suggested by the motions of projectiles, the parabolic path being treated as the undisturbed; and secondly, to the problem of determining the motions of a ternary or multiple system, with any laws of attraction or repulsion, and with one pre­dominant mass. This latter problem, which was touched upon in the former essay, is here resumed in a new manner, by forming and in­tegrating the differential equations of a new set of varying elements, entirely distinct in theory (though little differing in practice) from the elements conceived by Lagrange; and having this advantage, that the differentials of all the new elements for both the disturbed and disturbing masses may be expressed by the coefficients of one disturbing function.

VII. Second essay on a general method in dynamics

The former Essay contained a general method for reducing all the most important problems of dynamics to the study of one characteristic function, one central or ra­dical relation. It was remarked at the close of that Essay, that many eliminations required by this method in its first conception, might be avoided by a general trans­formation, introducing the time explicitly into a part S of the whole characteristic function V; and it is now proposed to fix the attention chiefly on this part S, and to call it the Principal Function . The properties of this part or function S, which were noticed briefly in the former Essay, are now more fully set forth; and especially its uses in questions of perturbation, in which it dispenses with many laborious and cir­cuitous processes, and enables us to express accurately the disturbed configuration of a system by the rules of undisturbed motion, if only the initial components of veloci­ties be changed in a suitable manner. Another manner of extending rigorously to disturbed motion the rules of undisturbed, by the gradual variation of elements, in number double the number of the coordinates or other marks of position of the system, which was first invented by Lagrange, and was afterwards improved by Poisson, is considered in this Second Essay under a form perhaps a little more general; and the general method of calculation which has already been applied to other analogous questions in optics and in dynamics by the author of the present Essay, is now applied to the integration of the equations which determine these ele­ments. This general method is founded chiefly on a combination of the principles of variations with those of partial differentials, and may furnish, when it shall be ma­tured by the labours of other analysts, a separate branch of algebra, which may be called perhaps the Calculus of Principal Functions ; because, in all the chief applica­tions of algebra to physics, and in a very extensive class of purely mathematical questions, it reduces the determination of many mutually connected functions to the search and study of one principal or central relation. When applied to the integration of the equations of varying elements, it suggests, as is now shown, the consideration of a certain Function of Elements , which may be variously chosen, and may either be rigorously determined, or at least approached to, with an indefinite accuracy, by a corollary of the general method. And to illustrate all these new general processes, but especially those which are connected with problems of perturbation, they are applied in this Essay to a very simple example, suggested by the motions of projectiles, the parabolic path being treated as the undisturbed. As a more important example, the problem of determining the motions of a ternary or multiple system, with any laws of attraction or repulsion, and with one predominant mass, which was touched upon in the former Essay, is here resumed in a new way, by forming and inte­grating the differential equations of a new set of varying elements, entirely distinct in theory (though little differing in practice) from the elements conceived by Lagrange, and having this advantage, that the differentials of all the new elements for both the disturbed and disturbing masses may be expressed by the coefficients of one disturbing function.