1 Euclidean Geometry
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Swimming in Spacetime: Motion by Cyclic Changes in Body Shape
RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing by ⌬ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding fixed, then increasing by (1). -
Foundations of Newtonian Dynamics: an Axiomatic Approach For
Foundations of Newtonian Dynamics: 1 An Axiomatic Approach for the Thinking Student C. J. Papachristou 2 Department of Physical Sciences, Hellenic Naval Academy, Piraeus 18539, Greece Abstract. Despite its apparent simplicity, Newtonian mechanics contains conceptual subtleties that may cause some confusion to the deep-thinking student. These subtle- ties concern fundamental issues such as, e.g., the number of independent laws needed to formulate the theory, or, the distinction between genuine physical laws and deriva- tive theorems. This article attempts to clarify these issues for the benefit of the stu- dent by revisiting the foundations of Newtonian dynamics and by proposing a rigor- ous axiomatic approach to the subject. This theoretical scheme is built upon two fun- damental postulates, namely, conservation of momentum and superposition property for interactions. Newton’s laws, as well as all familiar theorems of mechanics, are shown to follow from these basic principles. 1. Introduction Teaching introductory mechanics can be a major challenge, especially in a class of students that are not willing to take anything for granted! The problem is that, even some of the most prestigious textbooks on the subject may leave the student with some degree of confusion, which manifests itself in questions like the following: • Is the law of inertia (Newton’s first law) a law of motion (of free bodies) or is it a statement of existence (of inertial reference frames)? • Are the first two of Newton’s laws independent of each other? It appears that -
Higher Dimensional Conundra
Higher Dimensional Conundra Steven G. Krantz1 Abstract: In recent years, especially in the subject of harmonic analysis, there has been interest in geometric phenomena of RN as N → +∞. In the present paper we examine several spe- cific geometric phenomena in Euclidean space and calculate the asymptotics as the dimension gets large. 0 Introduction Typically when we do geometry we concentrate on a specific venue in a particular space. Often the context is Euclidean space, and often the work is done in R2 or R3. But in modern work there are many aspects of analysis that are linked to concrete aspects of geometry. And there is often interest in rendering the ideas in Hilbert space or some other infinite dimensional setting. Thus we want to see how the finite-dimensional result in RN changes as N → +∞. In the present paper we study some particular aspects of the geometry of RN and their asymptotic behavior as N →∞. We choose these particular examples because the results are surprising or especially interesting. We may hope that they will lead to further studies. It is a pleasure to thank Richard W. Cottle for a careful reading of an early draft of this paper and for useful comments. 1 Volume in RN Let us begin by calculating the volume of the unit ball in RN and the surface area of its bounding unit sphere. We let ΩN denote the former and ωN−1 denote the latter. In addition, we let Γ(x) be the celebrated Gamma function of L. Euler. It is a helpful intuition (which is literally true when x is an 1We are happy to thank the American Institute of Mathematics for its hospitality and support during this work. -
Phoronomy: Space, Construction, and Mathematizing Motion Marius Stan
To appear in Bennett McNulty (ed.), Kant’s Metaphysical Foundations of Natural Science: A Critical Guide. Cambridge University Press. Phoronomy: space, construction, and mathematizing motion Marius Stan With his chapter, Phoronomy, Kant defies even the seasoned interpreter of his philosophy of physics.1 Exegetes have given it little attention, and un- derstandably so: his aims are opaque, his turns in argument little motivated, and his context mysterious, which makes his project there look alienating. I seek to illuminate here some of the darker corners in that chapter. Specifi- cally, I aim to clarify three notions in it: his concepts of velocity, of compo- site motion, and of the construction required to compose motions. I defend three theses about Kant. 1) his choice of velocity concept is ul- timately insufficient. 2) he sided with the rationalist faction in the early- modern debate on directed quantities. 3) it remains an open question if his algebra of motion is a priori, though he believed it was. I begin in § 1 by explaining Kant’s notion of phoronomy and its argu- ment structure in his chapter. In § 2, I present four pictures of velocity cur- rent in Kant’s century, and I assess the one he chose. My § 3 is in three parts: a historical account of why algebra of motion became a topic of early modern debate; a synopsis of the two sides that emerged then; and a brief account of his contribution to the debate. Finally, § 4 assesses how general his account of composite motion is, and if it counts as a priori knowledge. -
Properties of Euclidean Space
Section 3.1 Properties of Euclidean Space As has been our convention throughout this course, we use the notation R2 to refer to the plane (two dimensional space); R3 for three dimensional space; and Rn to indicate n dimensional space. Alternatively, these spaces are often referred to as Euclidean spaces; for example, \three dimensional Euclidean space" refers to R3. In this section, we will point out many of the special features common to the Euclidean spaces; in Chapter 4, we will see that many of these features are shared by other \spaces" as well. Many of the graphics in this section are drawn in two dimensions (largely because this is the easiest space to visualize on paper), but you should be aware that the ideas presented apply to n dimensional Euclidean space, not just to two dimensional space. Vectors in Euclidean Space When we refer to a vector in Euclidean space, we mean directed line segments that are embedded in the space, such as the vector pictured below in R2: We often refer to a vector, such as the vector AB shown below, by its initial and terminal points; in this example, A is the initial point of AB, and B is the terminal point of the vector. While we often think of n dimensional space as being made up of points, we may equivalently consider it to be made up of vectors by identifying points and vectors. For instance, we identify the point (2; 1) in two dimensional Euclidean space with the vector with initial point (0; 0) and terminal point (2; 1): 1 Section 3.1 In this way, we think of n dimensional Euclidean space as being made up of n dimensional vectors. -
About Symmetries in Physics
LYCEN 9754 December 1997 ABOUT SYMMETRIES IN PHYSICS Dedicated to H. Reeh and R. Stora1 Fran¸cois Gieres Institut de Physique Nucl´eaire de Lyon, IN2P3/CNRS, Universit´eClaude Bernard 43, boulevard du 11 novembre 1918, F - 69622 - Villeurbanne CEDEX Abstract. The goal of this introduction to symmetries is to present some general ideas, to outline the fundamental concepts and results of the subject and to situate a bit the following arXiv:hep-th/9712154v1 16 Dec 1997 lectures of this school. [These notes represent the write-up of a lecture presented at the fifth S´eminaire Rhodanien de Physique “Sur les Sym´etries en Physique” held at Dolomieu (France), 17-21 March 1997. Up to the appendix and the graphics, it is to be published in Symmetries in Physics, F. Gieres, M. Kibler, C. Lucchesi and O. Piguet, eds. (Editions Fronti`eres, 1998).] 1I wish to dedicate these notes to my diploma and Ph.D. supervisors H. Reeh and R. Stora who devoted a major part of their scientific work to the understanding, description and explo- ration of symmetries in physics. Contents 1 Introduction ................................................... .......1 2 Symmetries of geometric objects ...................................2 3 Symmetries of the laws of nature ..................................5 1 Geometric (space-time) symmetries .............................6 2 Internal symmetries .............................................10 3 From global to local symmetries ...............................11 4 Combining geometric and internal symmetries ...............14 -
A Partition Property of Simplices in Euclidean Space
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number I, January 1990 A PARTITION PROPERTY OF SIMPLICES IN EUCLIDEAN SPACE P. FRANKL AND V. RODL 1. INTRODUCTION AND STATEMENT OF THE RESULTS Let ]Rn denote n-dimensional Euclidean space endowed with the standard metric. For x, y E ]Rn , their distance is denoted by Ix - yl. Definition 1.1 [E]. A subset B of ]Rd is called Ramsey if for every r ~ 2 there exists some n = n(r, B) with the following partition property. For every partition ]Rn = V; u ... U ~ , there exists some j, 1::::; j ::::; r, and Bj C fj such that B is congruent to Bj • In a series of papers, Erdos et al. [E] have investigated this property. They have shown that all Ramsey sets are spherical, that is, every Ramsey set is con- tained in an appropriate sphere. On the other hand, they have shown that the vertex set (and, therefore, all its subsets) of bricks (d-dimensional paral- lelepipeds) is Ramsey. The simplest sets that are spherical but cannot be embedded into the vertex set of a brick are the sets of obtuse triangles. In [FR 1], it is shown that they are indeed Ramsey, using Ramsey's Theorem (cf. [G2]) and the Product Theorem of [E]. The aim of the present paper is twofold. First, we want to show that the vertex set of every nondegenerate simplex in any dimension is Ramsey. Second, we want to show that for both simplices and bricks, and even for their products, one can in fact choose n(r, B) = c(B)logr, where c(B) is an appropriate positive constant. -
Euclidean Space - Wikipedia, the Free Encyclopedia Page 1 of 5
Euclidean space - Wikipedia, the free encyclopedia Page 1 of 5 Euclidean space From Wikipedia, the free encyclopedia In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria. Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. This approach brings the tools of algebra and calculus to bear on questions of geometry, and Every point in three-dimensional has the advantage that it generalizes easily to Euclidean Euclidean space is determined by three spaces of more than three dimensions. coordinates. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is the real coordinate space with three or more real number coordinates. Thus a point in Euclidean space is a tuple of real numbers, and distances are defined using the Euclidean distance formula. Mathematicians often denote the n-dimensional Euclidean space by , or sometimes if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension. Contents 1 Intuitive overview 2 Real coordinate space 3 Euclidean structure 4 Topology of Euclidean space 5 Generalizations 6 See also 7 References Intuitive overview One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. -
Central Limit Theorems for the Brownian Motion on Large Unitary Groups
CENTRAL LIMIT THEOREMS FOR THE BROWNIAN MOTION ON LARGE UNITARY GROUPS FLORENT BENAYCH-GEORGES Abstract. In this paper, we are concerned with the large n limit of the distributions of linear combinations of the entries of a Brownian motion on the group of n × n unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distributions are considered, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a very short proof of the asymptotic Gaussian feature of the entries of Haar distributed random unitary matrices, a result already proved by Diaconis et al. Abstract. Dans cet article, on consid`erela loi limite, lorsque n tend vers l’infini, de combinaisons lin´eairesdes coefficients d'un mouvement Brownien sur le groupe des matri- ces unitaires n × n. On prouve que le processus d'une telle combinaison lin´eaireconverge vers un processus gaussien. Diff´erentes ´echelles de temps et diff´erentes lois initiales sont consid´er´ees,donnant lieu `aplusieurs processus limites, li´es`ala construction g´eom´etrique du mouvement Brownien unitaire. En application, on propose une preuve tr`escourte du caract`ereasymptotiquement gaussien des coefficients d'une matrice unitaire distribu´ee selon la mesure de Haar, un r´esultatd´ej`aprouv´epar Diaconis et al. Introduction There is a natural definition of Brownian motion on any compact Lie group, whose distribution is sometimes called the heat kernel measure. Mainly due to its relations with the object from free probability theory called the free unitary Brownian motion and with the two-dimentional Yang-Mills theory, the Brownian motion on large unitary groups has appeared in several papers during the last decade. -
Rigid Motion – a Transformation That Preserves Distance and Angle Measure (The Shapes Are Congruent, Angles Are Congruent)
REVIEW OF TRANSFORMATIONS GEOMETRY Rigid motion – A transformation that preserves distance and angle measure (the shapes are congruent, angles are congruent). Isometry – A transformation that preserves distance (the shapes are congruent). Direct Isometry – A transformation that preserves orientation, the order of the lettering (ABCD) in the figure and the image are the same, either both clockwise or both counterclockwise (the shapes are congruent). Opposite Isometry – A transformation that DOES NOT preserve orientation, the order of the lettering (ABCD) is reversed, either clockwise or counterclockwise becomes clockwise (the shapes are congruent). Composition of transformations – is a combination of 2 or more transformations. In a composition, you perform each transformation on the image of the preceding transformation. Example: rRx axis O,180 , the little circle tells us that this is a composition of transformations, we also execute the transformations from right to left (backwards). If you would like a visual of this information or if you would like to quiz yourself, go to http://www.mathsisfun.com/geometry/transformations.html. Line Reflection Point Reflection Translations (Shift) Rotations (Turn) Glide Reflection Dilations (Multiply) Rigid Motion Rigid Motion Rigid Motion Rigid Motion Rigid Motion Not a rigid motion Opposite isometry Direct isometry Direct isometry Direct isometry Opposite isometry NOT an isometry Reverse orientation Same orientation Same orientation Same orientation Reverse orientation Same orientation Properties preserved: Properties preserved: Properties preserved: Properties preserved: Properties preserved: Properties preserved: 1. Distance 1. Distance 1. Distance 1. Distance 1. Distance 1. Angle measure 2. Angle measure 2. Angle measure 2. Angle measure 2. Angle measure 2. Angle measure 2. -
Physics, Philosophy, and the Foundations of Geometry Michael Friedman
Ditilogos 19 (2002), pp. 121-143. PHYSICS, PHILOSOPHY, AND THE FOUNDATIONS OF GEOMETRY MICHAEL FRIEDMAN Twentieth century philosophy of geometry is a development and continuation, of sorts, of late nineteenth century work on the mathematical foundations of geometry. Reichenbach, Schlick, a nd Carnap appealed to the earlier work of Riemann, Helmholtz, Poincare, and Hilbert, in particular, in articulating their new view of the nature and character of geometrical knowledge. This view took its starting point from a rejection of Kant's conception of the synthetic a priori status of specifically Euclidean geometry-and, indeed, from a rejection of any role at all for spatial intuition within pure mathematics. We must sharply distinguish, accordingly, between pure or mathematical geometry, which is an essentially uninterpreted axiomatic system making no reference whatsoever to spatial intuition or any other kind of extra axiomatic or extra-formal content, and applied or physical geometry, which then attempts to coordinate such an uninterpreted formal sys tern with some domain of physical facts given by experience. Since, however, there is always an optional element of decision in setting up such a coordination in the first place (we may coordinate the uninterpreted terms of purely formal geometry with light rays, stretched strings, rigid bodies, or whatever), the question of the geometry of physical space is not a purely empirical one but rather essentially involves an irreducibly conventional and in some sense arbitrary choice. In the end, there is therefore no fact of the matter whether physical space is E uclidean or non-Euclidean, except relative to one or another essentially arbitrary stipulation coordinating the uninterpreted terms of pure mathematical geometry with some or another empirical physical phenomenon. -
Arxiv:1704.03334V1 [Physics.Gen-Ph] 8 Apr 2017 Oin Ffltsaeadtime
WHAT DO WE KNOW ABOUT THE GEOMETRY OF SPACE? B. E. EICHINGER Department of Chemistry, University of Washington, Seattle, Washington 98195-1700 Abstract. The belief that three dimensional space is infinite and flat in the absence of matter is a canon of physics that has been in place since the time of Newton. The assumption that space is flat at infinity has guided several modern physical theories. But what do we actually know to support this belief? A simple argument, called the ”Telescope Principle”, asserts that all that we can know about space is bounded by observations. Physical theories are best when they can be verified by observations, and that should also apply to the geometry of space. The Telescope Principle is simple to state, but it leads to very interesting insights into relativity and Yang-Mills theory via projective equivalences of their respective spaces. 1. Newton and the Euclidean Background Newton asserted the existence of an Absolute Space which is infinite, three dimensional, and Euclidean.[1] This is a remarkable statement. How could Newton know anything about the nature of space at infinity? Obviously, he could not know what space is like at infinity, so what motivated this assertion (apart from Newton’s desire to make space the sensorium of an infinite God)? Perhaps it was that the geometric tools available to him at the time were restricted to the principles of Euclidean plane geometry and its extension to three dimensions, in which infinite space is inferred from the parallel postulate. Given these limited mathematical resources, there was really no other choice than Euclid for a description of the geometry of space within which to formulate a theory of motion of material bodies.