DOCSLIB.ORG

Newtonian Mechanics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Newtonian Mechanics Part II. Newtonian mechanics 30 4. Newtonian mechanics In this part of the book, we consider N-particle Newtonian mechanics: this chapter in- troduces the theory and its symmetries, the next discusses the rationale for regarding symmetry-related models as physically equivalent, and the one after that considers how we might seek to revise the theory to incorporate that judgment of physical equiv- alence. 4.1. Introduction to N-particle Newtonian mechanics In the first instance, we will describe this theory in terms of coordinates. We therefore expect that at least some of the structure of the theory will be unphysical, a mere ‘arte- fact of the coordinate system’. However, we will refrain from making intuitive judg- ments about which structural features correspond to physical features: in due course, we will use symmetry considerations to make such judgments. To set this up, we introduce a gadget that we will use repeatedly in what follows. Given some mathematical structure ⌦, an ⌦-valued variable is one whose intended range is ⌦—thus, given a Tarski-model A, a first-order variable is a A -valued variable, and | | a second-order variable of arity n is a ( A n)-valued variable. Then, given a set P | | ⇠ ,...,⇠ of ⌦-valued variables, the value space associated with that set consists of { 1 m} all maps from ⇠ ,...,⇠ to ⌦. Note that given an ordering on the variables, such a { 1 m} map can also be thought of as an m-tuple of elements of ⌦: such an m-tuple, after all, is simply a map from 1, . , m to ⌦. Thus, the value space is isomorphic to ⌦m.1 { } Now, without worrying too much about what doing this might mean, we take as given a coordinate system with x-, y- and z-axes, which persists over time; and we take as given some clock that measures the passage of time. We introduce one R- valued variable ptq, representing time, and three R-valued variables px1q, px2q and px3q, representing (respectively) the x-, y- and z-components of position. From these, 1Of course, one of the lessons that should have been taken from the previous chapters is that this kind of loose use of terms like ‘isomorphic’ is to be deplored. Consider this evidence that one should do as I say, not as I do. 31 we form the value spaces T (isomorphic to R) and X (isomorphic to R3); they repre- sent time and (position) space. We then introduce X-valued variables px1q,...,pxNq, with xn representing the position of the nth particle. The value-space for these variables will be denoted Q, and represents ‘configuration space’. Q therefore consists of maps from x ,...,x to X. Each such map is equivalent (via uncurrying) to a map from { 1 N} x ,...,x x1, x2, x3 R: we abbreviate the pair x , xi as xi , so that Q can also { 1 N}⇥{ }! h n i n be thought of as the value-space for 3N real-valued variables xi (with 1 i 3 and n 1 n N). The dynamics of the theory are given by the following set of differential equations (where 1 i 3 and 1 n N): d2xi m n = Fi (4.1) n dt n i Each equation in this set contains two new symbols, pmnq and pFnq; these are also taken to be real-valued (in the case of mn, to be positive real-valued). mn represents the mass i of the nth particle, and Fn the ith component of the force on the nth particle. A kinematically possible model for this theory will specify (constant) values for the mn, and the value of all xi and Fi at each t T; so the data specified consists of N real n n 2 numbers, and 6N functions from T to R. Such a kinematically possible model is a dynamically possible model if it is a solution of (4.1): that is, if for all t T, and all 1 i 3 2 and 1 n N, d2xi m n (t)=Fi (t) (4.2) n dt2 n This theory is something of a framework; we can make it more specific by adding i force-laws for the forces Fn. For example, if we suppose that each of our N particles has an electric charge q , and that they are in some electrical field given by Ei : X R3, n ! then we have i i Fn = qnE (xn) (4.3) On the other hand, if we are instead considering a theory where the N particles are mutually interacting through gravitation, then Gm m xi xi i p n n − p Fn = Â 2 (4.4) p=n xn xp xn xp 6 | − | | − | where x x := (xj xj )2 (4.5) | n − p| Â n − p s j 32 Finally, in the trivial case of free particles, the force-functions are especially simple: i Fn = 0 (4.6) 4.2. Changes of variables The theory above is (by design) stated for a single coordinate system. It is taken as read that the theory describes phenomena which are independent of a coordinate sys- tem. However, simply because the phenomena are independent of a coordinate system, it does not follow that the theory is: in particular, just because the theory holds in our coordinate system it does not follow that it will hold in some other coordinate system. What follows is merely that some other theory, systematically related to this one, will be true in that coordinate system. In other words, in order to make use of another co- ordinate system, we must specify how to translate the theory into a theory appropriate to that coordinate system. It will be best to illustrate this by an example. Let N = 1, and suppress the third 1 2 dimension: then, writing x1 = x and x1 = y, the theory reduces to the form mx¨ = Fx(x, y) (4.7a) my¨ = Fy(x, y) (4.7b) where dots indicate differentiation with respect to t. Writing the value-space of the set x , y as X Y (which is isomorphic to R2), we may consider solutions to this {p q p q} ⇥ theory as functions from T to X Y. ⇥ We now introduce new variables (the polar coordinates) prq and p✓q, and stipulate that pxq and pyq are translated into these new variables according to x r cos ✓ (4.8a) ⌘ y r sin ✓ (4.8b) ⌘ where have used the notation introduced in Chapter 2 (anticipating that this translation will be invertible). The value-space for the set of variables r , ✓ will be denoted R ⇥. We saw in {p q p q} ⇥ Chapter 2 that any translation induces a dual map on models; here, this corresponds to the fact that a translation from the variables pxq, pyq to the variables prq, p✓q induces a 33 map from R ⇥ to X Y, namely: ⇥ ⇥ (r, ✓) (r cos ✓, r sin ✓) (4.9) 7! In order that this map be a bijection, we stipulate that r 0, that (r, ✓)=(r, ✓ + 2⇡), ≥ and that (0, ✓)=(0, 0).2 This means we can write down expressions for the inverse translations: r x2 + y2 (4.10a) ⌘ q 1 y ✓ tan− (4.10b) ⌘ x ⇣ ⌘ together with the requirement that if x = 0, then ✓ = ⇡/2. We therefore have an invertible pair of translations, and so we can seek to translate the theory (4.7) into the new coordinate system. First, we observe that applying the derivative operator twice to each side of the translations (4.8) yields x¨ (r¨ r✓˙2) cos ✓ (2r˙✓˙ + r✓¨) sin ✓ (4.11) ⌘ − − y¨ (r¨ r✓˙2) sin ✓ +(2r˙✓˙ + r✓¨) cos ✓ (4.12) ⌘ − If we substitute in these expressions, then we get a theory whose differential equations are m(r¨ r✓˙2) cos ✓ (2r˙✓˙ + r✓¨) sin ✓ = Fx (4.13a) − − m(r¨ r✓˙2) sin ✓ +(2r˙✓˙ + r✓¨) cos ✓ = Fy (4.13b) − However, this theory is—in a certain sense—only a partial translation, since we are still expressing the forces in terms of their components in the original coordinate system. Provided that is understood, of course, the theory is still appropriate as a translation, and captures the same content as the original; indeed, there might even be situations where using one coordinate system to describe positions and another to describe forces might be appropriate (although it is not so easy to think of an example). Nevertheless, one might argue that it is unsatisfactory, given that it makes use of two coordinate systems. For this reason, the label of ‘Newtonian mechanics in polar 2Geometrically, this means that R ⇥ has the structure of a half-cylinder that has been ‘pinched off’ at ⇥ one end. 34 coordinates’ is typically reserved for the theory m(r¨ r✓˙2)=Fr (4.14a) − m(2r˙✓˙ + r✓¨)=F✓ (4.14b) which may be obtained by extending the translation ⌧ to the force expressions, accord- ing to Fx Fr cos ✓ F✓ sin ✓ (4.15) ⌘ − Fy Fr sin ✓ + F✓ cos ✓ (4.16) ⌘ That is, if we substitute these expressions into (4.7), then after some algebraic manipu- lation we obtain the translated theory (4.14). As we would expect, if we do the reverse process, then we find that the translations (4.10) will convert the theory (4.14) into the theory (4.7)—provided, that is, that we translate the force-functions according to x y Fr Fx + Fy (4.17a) ⌘ x2 + y2 x2 + y2 x y F✓ p Fx p Fy (4.17b) ⌘ x2 + y2 − x2 + y2 p p Why are these the ‘correct’ ways of extending the coordinate translations to trans- lations of force-expressions? The standard answer appeals to geometric arguments, in particular the fact that force is a vector quantity; thus, the components of forces in a new coordinate system are determined by computing how the ‘coordinate vectors’—unit vectors directed along the coordinate axes—change under the move to a new coordi- nate system.
Load more

Recommended publications
  • Projective Geometry: a Short Introduction
    Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g.
    [Show full text]
  • 4 Jul 2018 Is Spacetime As Physical As Is Space?
    Is spacetime as physical as is space? Mayeul Arminjon Univ. Grenoble Alpes, CNRS, Grenoble INP, 3SR, F-38000 Grenoble, France Abstract Two questions are investigated by looking successively at classical me- chanics, special relativity, and relativistic gravity: first, how is space re- lated with spacetime? The proposed answer is that each given reference fluid, that is a congruence of reference trajectories, defines a physical space. The points of that space are formally defined to be the world lines of the congruence. That space can be endowed with a natural structure of 3-D differentiable manifold, thus giving rise to a simple notion of spatial tensor — namely, a tensor on the space manifold. The second question is: does the geometric structure of the spacetime determine the physics, in particular, does it determine its relativistic or preferred- frame character? We find that it does not, for different physics (either relativistic or not) may be defined on the same spacetime structure — and also, the same physics can be implemented on different spacetime structures. MSC: 70A05 [Mechanics of particles and systems: Axiomatics, founda- tions] 70B05 [Mechanics of particles and systems: Kinematics of a particle] 83A05 [Relativity and gravitational theory: Special relativity] 83D05 [Relativity and gravitational theory: Relativistic gravitational theories other than Einstein’s] Keywords: Affine space; classical mechanics; special relativity; relativis- arXiv:1807.01997v1 [gr-qc] 4 Jul 2018 tic gravity; reference fluid. 1 Introduction and Summary What is more “physical”: space or spacetime? Today, many physicists would vote for the second one. Whereas, still in 1905, spacetime was not a known concept. It has been since realized that, even in classical mechanics, space and time are coupled: already in the minimum sense that space does not exist 1 without time and vice-versa, and also through the Galileo transformation.
    [Show full text]
  • Affine and Euclidean Geometry Affine and Projective Geometry, E
    AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda CHAPTER II: AFFINE AND EUCLIDEAN GEOMETRY AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 1. AFFINE SPACE 1.1 Definition of affine space A real affine space is a triple (A; V; φ) where A is a set of points, V is a real vector space and φ: A × A −! V is a map verifying: 1. 8P 2 A and 8u 2 V there exists a unique Q 2 A such that φ(P; Q) = u: 2. φ(P; Q) + φ(Q; R) = φ(P; R) for everyP; Q; R 2 A. Notation. We will write φ(P; Q) = PQ. The elements contained on the set A are called points of A and we will say that V is the vector space associated to the affine space (A; V; φ). We define the dimension of the affine space (A; V; φ) as dim A = dim V: AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda Examples 1. Every vector space V is an affine space with associated vector space V . Indeed, in the triple (A; V; φ), A =V and the map φ is given by φ: A × A −! V; φ(u; v) = v − u: 2. According to the previous example, (R2; R2; φ) is an affine space of dimension 2, (R3; R3; φ) is an affine space of dimension 3. In general (Rn; Rn; φ) is an affine space of dimension n. 1.1.1 Properties of affine spaces Let (A; V; φ) be a real affine space. The following statemens hold: 1.
    [Show full text]
  • Chapter II. Affine Spaces
    II.1. Spaces 1 Chapter II. Affine Spaces II.1. Spaces Note. In this section, we define an affine space on a set X of points and a vector space T . In particular, we use affine spaces to define a tangent space to X at point x. In Section VII.2 we define manifolds on affine spaces by mapping open sets of the manifold (taken as a Hausdorff topological space) into the affine space. Ultimately, we think of a manifold as pieces of the affine space which are “bent and pasted together” (like paper mache). We also consider subspaces of an affine space and translations of subspaces. We conclude with a discussion of coordinates and “charts” on an affine space. Definition II.1.01. An affine space with vector space T is a nonempty set X of points and a vector valued map d : X × X → T called a difference function, such that for all x, y, z ∈ X: (A i) d(x, y) + d(y, z) = d(x, z), (A ii) the restricted map dx = d|{x}×X : {x} × X → T defined as mapping (x, y) 7→ d(x, y) is a bijection. We denote both the set and the affine space as X. II.1. Spaces 2 Note. We want to think of a vector as an arrow between two points (the “head” and “tail” of the vector). So d(x, y) is the vector from point x to point y. Then we see that (A i) is just the usual “parallelogram property” of the addition of vectors.
    [Show full text]
  • The Euler Characteristic of an Affine Space Form Is Zero by B
    BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 81, Number 5, September 1975 THE EULER CHARACTERISTIC OF AN AFFINE SPACE FORM IS ZERO BY B. KOST ANT AND D. SULLIVAN Communicated by Edgar H. Brown, Jr., May 27, 1975 An affine space form is a compact manifold obtained by forming the quo­ tient of ordinary space i?" by a discrete group of affine transformations. An af­ fine manifold is one which admits a covering by coordinate systems where the overlap transformations are restrictions of affine transformations on Rn. Affine space forms are characterized among affine manifolds by a completeness property of straight lines in the universal cover. If n — 2m is even, it is unknown whether or not the Euler characteristic of a compact affine manifold can be nonzero. The curvature definition of charac­ teristic classes shows that all classes, except possibly the Euler class, vanish. This argument breaks down for the Euler class because the pfaffian polynomials which would figure in the curvature argument is not invariant for Gl(nf R) but only for SO(n, R). We settle this question for the subclass of affine space forms by an argument analogous to the potential curvature argument. The point is that all the affine transformations x \—> Ax 4- b in the discrete group of the space form satisfy: 1 is an eigenvalue of A. For otherwise, the trans­ formation would have a fixed point. But then our theorem is proved by the fol­ lowing p THEOREM. If a representation G h-• Gl(n, R) satisfies 1 is an eigenvalue of p(g) for all g in G, then any vector bundle associated by p to a principal G-bundle has Euler characteristic zero.
    [Show full text]
  • A Modular Compactification of the General Linear Group
    Documenta Math. 553 A Modular Compactification of the General Linear Group Ivan Kausz Received: June 2, 2000 Communicated by Peter Schneider Abstract. We define a certain compactifiction of the general linear group and give a modular description for its points with values in arbi- trary schemes. This is a first step in the construction of a higher rank generalization of Gieseker's degeneration of moduli spaces of vector bundles over a curve. We show that our compactification has simi- lar properties as the \wonderful compactification” of algebraic groups of adjoint type as studied by de Concini and Procesi. As a byprod- uct we obtain a modular description of the points of the wonderful compactification of PGln. 1991 Mathematics Subject Classification: 14H60 14M15 20G Keywords and Phrases: moduli of vector bundles on curves, modular compactification, general linear group 1. Introduction In this paper we give a modular description of a certain compactification KGln of the general linear group Gln. The variety KGln is constructed as follows: First one embeds Gln in the obvious way in the projective space which contains the affine space of n × n matrices as a standard open set. Then one succes- sively blows up the closed subschemes defined by the vanishing of the r × r subminors (1 ≤ r ≤ n), along with the intersection of these subschemes with the hyperplane at infinity. We were led to the problem of finding a modular description of KGln in the course of our research on the degeneration of moduli spaces of vector bundles. Let me explain in some detail the relevance of compactifications of Gln in this context.
    [Show full text]
  • Chapter 2 Affine Algebraic Geometry
    Chapter 2 Affine Algebraic Geometry 2.1 The Algebraic-Geometric Dictionary The correspondence between algebra and geometry is closest in affine algebraic geom- etry, where the basic objects are solutions to systems of polynomial equations. For many applications, it suffices to work over the real R, or the complex numbers C. Since important applications such as coding theory or symbolic computation require finite fields, Fq , or the rational numbers, Q, we shall develop algebraic geometry over an arbitrary field, F, and keep in mind the important cases of R and C. For algebraically closed fields, there is an exact and easily motivated correspondence be- tween algebraic and geometric concepts. When the field is not algebraically closed, this correspondence weakens considerably. When that occurs, we will use the case of algebraically closed fields as our guide and base our definitions on algebra. Similarly, the strongest and most elegant results in algebraic geometry hold only for algebraically closed fields. We will invoke the hypothesis that F is algebraically closed to obtain these results, and then discuss what holds for arbitrary fields, par- ticularly the real numbers. Since many important varieties have structures which are independent of the field of definition, we feel this approach is justified—and it keeps our presentation elementary and motivated. Lastly, for the most part it will suffice to let F be R or C; not only are these the most important cases, but they are also the sources of our geometric intuitions. n Let A denote affine n-space over F. This is the set of all n-tuples (t1,...,tn) of elements of F.
    [Show full text]
  • Affine Spaces Within Projective Spaces
    Affine Spaces within Projective Spaces Andrea Blunck∗ Hans Havlicek Abstract We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finite- dimensional or pappian. Mathematics Subject Classification (1991): 51A30, 51A40, 51A45. Key Words: Affine space, regulus, dual spread. 1 Introduction The aim of this paper is to equip the set of complements of a distinguished subspace in an arbitrary projective space with the structure of an affine space. In this introductory section we first formulate the problem more exactly and then describe certain special cases that have already been treated in the literature. Let K be a not necessarily commutative field, and let V be a left vector space over K with arbitrary | not necessarily finite | dimension. Moreover, fix a subspace W of V . We are interested in the set := S W V = W S (1) S f ≤ j ⊕ g of all complements of W . Note that often we shall take the projective point of view; then we identify with the set S of all those projective subspaces of the projective space P(K; V ) that are complementary to the fixed projective subspace induced by W . In the special case that W is a hyperplane, the set obviously is the point set of an affine S space, namely, of the affine derivation of P(K; V ) w.r.t.
    [Show full text]
  • Statistical Manifold As an Affine Space
    ARTICLE IN PRESS Journal of Mathematical Psychology 50 (2006) 60–65 www.elsevier.com/locate/jmp Statistical manifold as an affine space: A functional equation approach Jun Zhanga,Ã, Peter Ha¨sto¨b,1 aDepartment of Psychology, 525 East University Ave., University of Michigan, Ann Arbor, MI 48109-1109, USA bDepartment of Mathematics and Statistics, P. O. Box 68, 00014 University of Helsinki, Finland Received 7 January 2005; received in revised form 9 July 2005 Available online 11 October 2005 Abstract A statistical manifold Mm consists of positive functions f such that fdm defines a probability measure. In order to define an atlas on the manifold, it is viewed as an affine space associated with a subspace of the Orlicz space LF. This leads to a functional equation whose solution, after imposing the linearity constrain in line with the vector space assumption, gives rise to a general form of mappings between the affine probability manifold and the vector (Orlicz) space. These results generalize the exponential statistical manifold and clarify some foundational issues in non-parametric information geometry. r 2005 Published by Elsevier Inc. Keywords: Probability density function; Exponential family; Non-parameteric; Information geometry 1. Introduction theory, stochastic process, neural computation, machine learning, Bayesian statistics, and other related fields (Amari, Information geometry investigates the differential geo- 1985; Amari & Nagaoka, 1993/2000). Recently, an interest in metric structure of the manifold of probability density the geometry associated with non-parametric probability functions with continuous support (or probability distribu- densities has arisen (Pistone & Sempi, 1995; Giblisco & tions with discrete support). Treating a family of parametric Pistone, 1998; Pistone & Rogantin, 1999; Grasselli, 2005).
    [Show full text]
  • THE HILBERT SCHEME of INFINITE AFFINE SPACE and ALGEBRAIC K-THEORY 3 Cohomology Theories and the Corresponding Transfers Is Given in [?, §1.1]
    THE HILBERT SCHEME OF INFINITE AFFINE SPACE AND ALGEBRAIC K-THEORY MARC HOYOIS, JOACHIM JELISIEJEW, DENIS NARDIN, BURT TOTARO, AND MARIA YAKERSON ∞ 1 Abstract. We study the Hilbert scheme Hilbd(A ) from an A -homotopical viewpoint and obtain appli- ∞ 1 cations to algebraic K-theory. We show that the Hilbert scheme Hilbd(A ) is A -equivalent to the Grass- ∞ 1 n mannian of (d − 1)-planes in A . We then describe the A -homotopy type of Hilbd(A ) in a range, for n n large compared to d. For example, we compute the integral cohomology of Hilbd(A )(C) in a range. We also deduce that the forgetful map FFlat → Vect from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an A1-equivalence after group completion. This implies that the moduli stack FFlat, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum kgl representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the kgl-homology of smooth proper schemes over a perfect field. Contents 1. Introduction 1 1 2. Thestacks FFlatd and Vectd−1 are A -equivalent 4 3. A1-equivalence between the group completions of the stacks FFlat and Vect 6 4. Consequences for the Hilbert scheme of A∞ 9 5. Theeffective motivic K-theory spectrum 14 6. Comparison with algebraic cobordism 17 7. The Hilbert scheme of finite lci schemes of degree 3 18 8. Stability theorems for the Hilbert scheme 20 References 25 1.
    [Show full text]
  • Two Examples of Affine Manifolds
    PACIFIC JOURNAL OF MATHEMATICS Vol. 94, No. 2, 1981 TWO EXAMPLES OF AFFINE MANIFOLDS WILLIAM M. GOLDMAN An affine manifold is a manifold with a distinguished system of affine coordinates, namely, an open covering by charts which map homeomorphically onto open sets in an affine space E such that on overlapping charts the homeo- morphisms differ by an affine automorphism of E. Some, but certainly not all, affine manifolds arise as quotients Ω/Γ of a domain in E by a discrete group Γ of affine trans- formations acting properly and freely. In that case we identify Ω with a covering space of the affine manifold. If Ω—E, then we say the affine manifold is complete. In general, however, there is only a local homeomorphism of the universal covering into E, which is equivariant with respect to a certain affine representation of the fundamental group. The image of this representation is a certain sub- group of the affine group on E, is called the affine holonomy and is well defined up to conjugacy in the affine group. See Fried, Goldman, and Hirsch. THEOREM. There exists a compact affine manifold M satisfying the following conditions: (1) M is the quotient of a proper convex domain which is not a convex cone. (2) M is incomplete and the affine holonomy group leaves in- variant no proper affine subspace. The example we give will be three-dimensional (it is a torus bundle over the circle); by taking products with compact complete manifolds, we obtain similar examples in all dimensions greater than two.
    [Show full text]
  • Lecture 4: Linear Algebra and Convexity
    Integer Programming ISE 418 Lecture 4 Dr. Ted Ralphs ISE 418 Lecture 4 1 Reading for This Lecture • N&W Sections I.4.1-I.4.3 1 ISE 418 Lecture 4 2 Linear Algebra Review: Linear Independence Definition 1. A finite collection of vectors x1; : : : ; xk 2 Rn is linearly Pk i independent if the unique solution to i=1 λix = 0 is λi = 0; i 2 [1::k]. Otherwise, the vectors are linearly dependent. Let A be a square matrix. Then, the following statements are equivalent: • The matrix A is invertible. • The matrix A> is invertible. • The determinant of A is nonzero. • The rows of A are linearly independent. • The columns of A are linearly independent. • For every vector b, the system Ax = b has a unique solution. • There exists some vector b for which the system Ax = b has a unique solution. 2 ISE 418 Lecture 4 3 Linear Algebra Review: Affine Independence Definition 2. A finite collection of vectors x1; : : : ; xk 2 Rn is affinely independent if the vectors x2 − x1; : : : ; xk − x1 2 Rn are linearly independent. • Linear independence implies affine independence, but not vice versa. • The property of linear independence is with respect to a given origin. • Affine independence is essentially a \coordinate-free" version of linear independence. Proposition 1. The following statements are equivalent: n 1. x1; : : : ; xk 2 R are affinely independent. 2. x2 − x1; : : : ; xk − x1 are linearly independent. n+1 3. (x1; 1);:::; (xk; 1) 2 R are linearly independent. 3 ISE 418 Lecture 4 4 Linear Algebra Review: Subspaces Definition 3.
    [Show full text]