DOCSLIB.ORG

Solutions Manual

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Solutions Manual SOLUTIONS MANUAL Introduction to Classical Mechanics With Problems and Solutions David Morin Cambridge University Press TO THE INSTRUCTOR: I have tried to pay as much attention to detail in these exercise solutions as I did in the problem solutions in the text. But despite working through each solution numerous times during the various stages of completion, there are bound to be errors. So please let me know if anything looks amiss. Also, to keep this pdf ¯le from escaping to the web, PLEASE don't distribute it to anyone, with the exception of your teaching assistants. And please make sure they also agree to this. Once this ¯le gets free, there's no going back. In addition to any comments you have on these solutions, I welcome any comments on the book in general. I hope you're enjoying using it! David Morin [email protected] (Version 2, April 2008) °c David Morin 2008 Chapter 1 Strategies for solving problems 1.8. Pendulum on the moon pThe only way to get units of time from `, g, and m is through the combination `=g. Therefore, p r TM `=gM gE p = p = =) TM ¼ 6 TE ¼ 7:3 s: (1) TE `=gE gM 1.9. Escape velocity (a) Using M = ½V , we have r 2G ¢ (4=3)¼R3½ p v = = (8=3)¼GR2½: (2) R p (b) We see that v / R ½. Therefore, p v RJ ½J 1 J = p = 11 ¢ p = 5:5: (3) vE RE ½E 4 1.10. Downhill projectile The angle ¯ is some function of the form, ¯ = f(θ; m; v0; g). In terms of units, we can write 1 = f(1; kg; m=s; m=s2). We can't have any m dependence, because there is nothing to cancel the kg. And we also can't have any v0 or g dependence, because they would have to appear in the ratio v0=g to cancel the meters, but then seconds would remain. Therefore, ¯ can depend on at most θ. (And it clearly does depend on θ, because ¯ = 90± for θ = 0 or 90±, but ¯ 6= 90± for θ 6= 0 or 90±.) 1.11. Waves on a string The speed v is some function of the form, v = f(M; L; T ). In terms of units, we can 2 write m=s = f(kg; m; kg m=s ). We needp to get rid of the kg's, so we must use the ratio T=M. We then quickly seep that LT=M has the correct units of m=s. Note that this can also be written as T=½, where ½ is the mass density per unit length. 1.12. Vibrating water drop The frequency º is some function of the form, º = f(R; ½; S). In terms of units, we 3 2 can write 1=s = f(m; kg=m ; kg=s ). Wep need to get rid of the kg's, so we must use 3 the ratio S=½. We then quickly seep that S=½R has the correct units of 1=s. Note that this can also be written as S=M, where M is the mass of the water drop. 1 2 CHAPTER 1. STRATEGIES FOR SOLVING PROBLEMS 1.13. Atwood's machine (a) This gives a1 = 0. (Half of m2 balances each of m1 and m3.) (b) Ignore the m2m3 terms, which gives a1 = ¡g. (Simply in freefall.) (c) Ignore the terms involving m1, which gives a1 = 3g.(m2 and m3 are in freefall. And for every meter they go down, a total of three meters of string appears above them, so m1 goes up three meters.) (d) Ignore the m1m3 terms, which gives a1 = g.(m2 goes down at g, and m1 and m3 go up at g.) (e) This gives a1 = ¡g=3. (Not obvious.) 1.14. Cone frustum The correct answer must reduce to the volume of a cylinder, ¼a2h, when a = b. Only the 2nd, 3rd, and 5th options satisfy this. The correct answer must also reduce to the volume of a cone, ¼b2h=3, when a = 0. Only the 1st, 3rd, and 4th options satisfy this. The correct answer must therefore be the 3rd one, ¼h(a2 + ab + b2)=3. 1.15. Landing at the corner The correct answer must go to in¯nity for θ ! 90±. Only the 2nd and 3rd options satisfy this. The correct answer must also go to in¯nity for θ ! 45±. Only the 1st and 2nd options satisfy this. The correct answer must therefore be the 2nd one. 1.16. Projectile with drag Using the Taylor series for e¡®t, we have ³ ´ ³ ´ 1 g 2 2 gt y(t) = v0 sin θ + 1 ¡ (1 ¡ ®t + ® t =2 ¡ ¢ ¢ ¢) ¡ ³® ´® ³ ´ ® g gt ¼ v sin θ + t ¡ ®t2=2 ¡ 0 ® ® 1 1 = (v sin θ)t ¡ gt2 ¡ ®t2v sin θ: (4) 0 2 2 0 If ® ¿ g=(v0 sin θ), then the third term is much smaller than the second, and we obtain the desired result. So ® ¿ g=(v0 sin θ) is what we mean by \small ®." However, we also assumed ®t ¿ 1 in the expansion for e¡®t above, so we should check that this doesn't necessitate a stricter upper bound on ®. And indeed, the total time of flight is less than 2v0 sin θ=g (because this t makes the above y(t) negative), so the condition ® ¿ g=(v0 sin θ) implies ®t ¿ (g=v0 sin θ)(2v0 sin θ=g) = 2. So ®t ¿ 1 is guaranteed by ® ¿ g=(v0 sin θ). 1.17. Pendulum Here is a Maple program that does the job: q:=3.14159/2: # initial θ value q1:=0: # initial θ speed e:=.0001: # a small time interval i:=0: # i will count the number of time steps while q>0 do # run the program while θ > 0 i:=i+1: # increase the counter by 1 q2:=-(9.8)*sin(q)/1: # the given equation q:=q+e*q1: # how q changes, by definition of q1 q1:=q1+e*q2: # how q1 changes, by definition of q2 end do: # the Maple command to stop the do loop i*e; # print the value of the time This yields a time of t = 0:5923 s. If we instead use a time interval of .00001 s, we obtain t = 0:59227 s. And a time interval of .000001 s gives t = 0:592263 s. 3 1.18. Distance with damping In thex Ä = ¡Ax_ case, we have the following Maple program: x:=0: # initial x value x1:=2: # initial x speed T:=1: # the total time e:=.001: # a small time interval for i to T/e do # run the program for a time T x2:=-(1)*x1: # the given equation x:=x+e*x1: # how x changes, by definition of x1 x1:=x1+e*x2: # how x1 changes, by definition of x2 end do: # the Maple command to stop the do loop x; # print the value of the position To run the program for di®erent times, we simply need to change the value of T in the 3rd line. Letting T equal 1 gives a ¯nal position of 1.264. Letting T equal 10 and 100 gives ¯nal positions of 1.99991 and 1.9999996, respectively. These approach 2. In thex Ä = ¡Ax_ 2 case, the only change in the entire program is in the 6th line, where we now have the square of x1: x2:=-(1)*x1^2: # the given equation Letting T equal 1, 10, 100, 1000, and 10000, gives ¯nal positions of 1.099, 3.044, 5.302, 7.600, and 9.903, respectively. Looking at the successive di®erences between these values, we see that they approach roughly 2.3. This constant di®erence for inputs of powers of 10 implies a log dependence on the time. 4 CHAPTER 1. STRATEGIES FOR SOLVING PROBLEMS Chapter 2 Statics 2.20. Block under an overhang Let's break up the forces into components parallel and perpendicular to the over- hang. Let positive Ff point up along the overhang. Balancing the forces parallel and perpendicular to the overhang gives, respectively, Ff = Mg sin ¯ + Mg cos ¯; and N = Mg sin ¯ ¡ Mg cos ¯: (5) N must be positive, so we immediately see that ¯ must be at least 45± if there is any chance that the setup is static. The coe±cient ¹ tells us that jFf j · ¹N. Using Eq. (5), this inequality becomes ¹ + 1 Mg(sin ¯ + cos ¯) · ¹Mg(sin ¯ ¡ cos ¯) =) · tan ¯: (6) ¹ ¡ 1 We see that we must have ¹ > 1 in order for there to exist any values of ¯ that satisfy this inequality. If ¹ ! 1, then ¯ can be as small as 45±, but it can't be any smaller. 2.21. Pulling a block The Fy forces tell us that N + F sin θ ¡ mg = 0 =) N = mg ¡ F sin θ. And assuming that the block slips, the Fx forces tell us that F cos θ > ¹N. Therefore, ¹mg F cos θ > ¹(mg ¡ F sin θ) =) F > : (7) cos θ + ¹ sin θ Taking the derivative top minimize this then gives tan θ = ¹. Plugging this θ back into F gives F > ¹mg= 1 + ¹2. If ¹ = 0, we have θ = 0 and F > 0. If ¹ ! 1, we have θ ¼ 90± and F > mg. 2.22. Holding a cone Let F be the friction force at each ¯nger. Then the Fy forces on the cone tell us that 2F cos θ ¡ 2N sin θ ¡ mg = 0. But F · ¹N. Therefore, mg 2¹N cos θ ¡ 2N sin θ ¡ mg > 0 =) N ¸ : (8) 2(¹ cos θ ¡ sin θ) This is the desired minimum normal force.
Load more

Recommended publications
  • [Math.DS] 18 Jan 2005 Complexity of Piecewise Convex Transformations
    Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards Eugene Gutkin ∗ and Serge Tabachnikov † November 12, 2018 Abstract ABSTRACT We introduce the class of piecewise convex transfor- mations, and study their complexity. We apply the results to the complexity of polygonal billiards on surfaces of constant curvature. Introduction The following situation frequently occurs in geometric dynamics. There is a phase space X, a transformation T : X → X; and there is a finite de- composition P : X = X(a) ∪ X(b) ∪ ··· . Let A = {a, b, . } be the corresponding alphabet. A phase point x ∈ X is regular if every element of the orbit x, T x, T 2x, . belongs to a unique atom of P. Suppose that arXiv:math/0412335v2 [math.DS] 18 Jan 2005 x ∈ X(a), T x ∈ X(b), etc. The corresponding word a b ··· is the code of x. Let Σ(n) be the set of words in A of length n obtained by coding points in X. The positive function f(n) = |Σ(n)| is the associated complexity. Its ∗UCLA, Math. Dpt., Los Angeles, CA 90095, USA and IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro RJ, Brasil. E-mail: [email protected],[email protected] †Department of Mathematics, Pennsylvania State University, University Park, PA 16802. E-mail: [email protected] 1 behavior as n →∞1 is an important characteristic of the dynamical system in question. The following examples have motivated our study. Example A. Let P ⊂ R2 be a polygon with sides a, b, . , and let X be the phase space of the billiard map Tb in P .
    [Show full text]
  • © 2016 Ki Yeun Kim DYNAMICS of BOUNCING RIGID BODIES and BILLIARDS in the SPACES of CONSTANT CURVATURE
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Illinois Digital Environment for Access to Learning and Scholarship Repository © 2016 Ki Yeun Kim DYNAMICS OF BOUNCING RIGID BODIES AND BILLIARDS IN THE SPACES OF CONSTANT CURVATURE BY KI YEUN KIM DISSERTATION Submied in partial fulllment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2016 Urbana, Illinois Doctoral Commiee: Professor Yuliy Baryshnikov, Chair Professor Vadim Zharnitsky, Director of Research Associate Professor Robert DeVille Associate Professor Zoi Rapti Abstract Mathematical billiard is a dynamical system studying the motion of a mass point inside a domain. e point moves along a straight line in the domain and makes specular reections at the boundary. e theory of billiards has developed extensively for itself and for further applications. For example, billiards serve as natural models to many systems involving elastic collisions. One notable example is the system of spherical gas particles, which can be described as a billiard on a higher dimensional space with a semi-dispersing boundary. In the rst part of this dissertation, we study the collisions of a two-dimensional rigid body using billiard dynamics. We rst dene a dumbbell system, which consists of two point masses connected by a weightless rod. We assume the dumbbell moves freely in the air and makes elastic collisions at a at boundary. For arbitrary mass choices, we use billiard techniques to nd the sharp bound on the number of collisions of the dumbbell system in terms of the mass ratio.
    [Show full text]
  • Complexity of Piecewise Convex Transformations in Two Dimensions, with Applications to Polygonal Billiards on Surfaces of Constant Curvature
    MOSCOW MATHEMATICAL JOURNAL Volume 6, Number 4, October–December 2006, Pages 673–701 COMPLEXITY OF PIECEWISE CONVEX TRANSFORMATIONS IN TWO DIMENSIONS, WITH APPLICATIONS TO POLYGONAL BILLIARDS ON SURFACES OF CONSTANT CURVATURE EUGENE GUTKIN AND SERGE TABACHNIKOV To A. A. Kirillov for his 70th birthday Abstract. We introduce piecewise convex transformations, and de- velop geometric tools to study their complexity. We apply the results to the complexity of polygonal inner and outer billiards on surfaces of constant curvature. 2000 Math. Subj. Class. 53D25, 37E99, 37B10. Key words and phrases. Geodesic polygon, constant curvature, complexity, inner billiard, outer billiard. Introduction Transformations that arise in geometric dynamics often fit into the following scheme. The phase space of the transformation, T : X → X, has a finite decompo- sition P : X = Xa ∪ Xb ∪··· into “nice” subsets with respect to a geometric struc- ture on X. (For instance, Xa, Xb, ... are convex.) The restrictions T : Xa → X, T : Xb → X, ... preserve that structure. The interiors of Xa, Xb, ... are dis- joint, and T is discontinuous (or not defined) on the union of their boundaries, −1 −1 ∂P = ∂Xa ∪ ∂Xb ∪··· . Set P1 = P. The sets T (Xa) ∩ Xa, T (Xa) ∩ Xb, ... form the decomposition P2, which is a refinement of P1; it provides a defining partition for the transformation T 2. Iterating this process, we obtain a tower of n decompositions Pn, n > 1, where Pn plays for T the same role that P played for T . Let A = {a, b, ... } be the alphabet labeling the atoms of P. A phase point x ∈ X is regular if every point of the orbit x, Tx, T 2x, ..
    [Show full text]
  • Outer Billiards Outside a Regular Octagon: Periodicity of Almost All Orbits and Existence of an Aperiodic Orbit F
    ISSN 1064-5624, Doklady Mathematics, 2018, Vol. 98, No. 1, pp. 1–4. © Pleiades Publishing, Ltd., 2018. Original Russian Text © F.D. Rukhovich, 2018, published in Doklady Akademii Nauk, 2018, Vol. 481, No. 3, pp. 00000–00000. MATHEMATICS Outer Billiards outside a Regular Octagon: Periodicity of Almost All Orbits and Existence of an Aperiodic Orbit F. D. Rukhovich Presented by Academician of the RAS A.L. Semenov November 30, 2017 Received November 30, 2017 Abstract—The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Addi- tionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed. DOI: 10.1134/S1064562418050101 Let be a convex shape and p be a point outside it. Tabachnikov [1] has solved the periodicity prob- Consider the right (with respect to p) tangent to . lems in the case of regular 3-, 4-, and 6-gons. Specifi- LetTp T() p be the point symmetric to p with cally, there is no aperiodic point for them, as well as for respect to the tangency point. 5-gons, for which aperiodic points exist, but have Definition 1. The mapping T is called an outer bil- measure zero. In his 2005 monograph [6], Tabach- liard, and is called an outer billiard table. nikov presented computer simulation results for an octagon, but noted that a rigor analysis is still not Definition 2. A point p outside is said to be peri- available and there are no results for other cases. Later, odic if there exists a positive integer n such that a regular pentagon and associated symbolic dynamics Tpn p.
    [Show full text]
  • Recent Advances in Open Billiards with Some Open Problems Carl P
    Recent advances in open billiards with some open problems Carl P. Dettmann, Department of Mathematics, University of Bristol, UK Abstract Much recent interest has focused on \open" dynamical systems, in which a classical map or flow is considered only until the trajectory reaches a \hole", at which the dynamics is no longer considered. Here we consider questions pertaining to the survival probability as a function of time, given an initial measure on phase space. We focus on the case of billiard dynamics, namely that of a point particle moving with constant velocity except for mirror-like reflections at the boundary, and give a number of recent results, physical applications and open problems. 1 Introduction A mathematical billiard is a dynamical system in which a point particle moves with constant speed in a straight line in a compact domain D ⊂ Rd with a piecewise smooth1 boundary @D and making mirror-like2 reflections whenever it reaches the boundary. We can assume that the speed and mass are both equal to unity. In some cases it is convenient to use periodic boundary conditions with obstacles in the interior, so Rd is replaced by the torus Td. Here we will mostly consider the planar case d = 2. Billiards are of interest in mathematics as examples of many different dynamical behaviours (see Fig. 1 below) and in physics as models in statistical mechanics and the classical limit of waves moving in homogeneous cavities; more details for both mathematics and physics are given below. An effort has been made to keep the discussion as non-technical and self-contained as possible; for further definitions and discussion, please see [20, 28, 37].
    [Show full text]
  • A Brief Overview of Outer Billiards on Polygons
    Degree project A Brief Overview of Outer Billiards on Polygons Author: Michelle Zunkovic Supervisor: Hans Frisk Examiner: Karl-Olof Lindahl Date: 2015-12-17 Course Code: 2MA11E Subject:Mathematics Level: Bachelor Department Of Mathematics A Brief Overview of Outer Billiards on Polygons Michelle Zunkovic December 17, 2015 Contents 1 Introduction 3 2 Theory 3 2.1 Affine transformations . .3 2.2 Definition of Outer Billiards . .4 2.3 Boundedness and unboundedness of orbits . .4 2.4 Periodic orbits . .6 2.5 The T 2 map and necklace dynamics . .6 2.6 Quasi rational billiards . .8 3 Some different polygons 8 3.1 Orbits of the 2-gon . .8 3.2 Orbits of the triangle . .9 3.3 Orbits of the quadrilaterals . 10 3.3.1 The square . 10 3.3.2 Trapezoids and kites . 10 3.4 The regular pentagon . 11 3.5 The regular septagon . 12 3.6 Structuring all polygons . 13 4 Double kite 13 4.1 Background . 13 4.2 Method . 15 4.3 Result . 15 5 Conclusions 16 2 Abstract Outer billiards were presented by J¨urgenMoser in 1978 as a toy model of the solar system. It is a geometric construction concerning the motions around a convex shaped space. We are going to bring up the basic ideas with many figures and not focus on the proofs. Explanations how different types of orbits behave are given. 1 Introduction It was J¨urgenMoser that aroused the interest of outer billiards to the public. In 1978 Moser published an article named "Is the Solar System Stable?" [1].
    [Show full text]
  • Outer Billiard Outside Regular Polygons Nicolas Bedaride, Julien Cassaigne
    Outer billiard outside regular polygons Nicolas Bedaride, Julien Cassaigne To cite this version: Nicolas Bedaride, Julien Cassaigne. Outer billiard outside regular polygons. Journal of the London Mathematical Society, London Mathematical Society, 2011, 10.1112/jlms/jdr010. hal-01219092 HAL Id: hal-01219092 https://hal.archives-ouvertes.fr/hal-01219092 Submitted on 22 Oct 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Outer billiard outside regular polygons Nicolas Bedaride∗ Julien Cassaigney ABSTRACT We consider the outer billiard outside a regular convex polygon. We deal with the case of regular polygons with 3; 4; 5; 6 or 10 sides. We de- scribe the symbolic dynamics of the map and compute the complexity of the language. Keywords: Symbolic dynamic, outer billiard, complexity function, words. AMS codes: 37A35 ; 37C35; 05A16; 28D20. 1 Introduction An outer billiard table is a compact convex domain P . Pick a point M outside P . There are two half-lines starting from M tangent to P , choose one of them, say the one to the right from the view-point of M, and reflect M with respect to the tangency point. One obtains a new point, N, and the transformation T : TM = N is the outer (a.k.a.
    [Show full text]
  • Outer Billiards with Contraction: Attracting Cantor Sets
    Outer Billiards with Contraction: Attracting Cantor Sets In-Jee Jeong September 27, 2018 Abstract We consider the outer billiards map with contraction outside polygons. We construct a 1-parameter family of systems such that each system has an open set in which the dynamics is reduced to that of a piecewise con- traction on the interval. Using the theory of rotation numbers, we deduce that every point inside the open set is asymptotic to either a single peri- odic orbit (rational case) or a Cantor set (irrational case). In particular, we deduce existence of an attracting Cantor set for certain parameter val- ues. Moreover, for a different choice of a 1-parameter family, we prove that the system is uniquely ergodic; in particular, the entire domain is asymptotic to a single attractor. 1 Introduction Let P be a convex polygon in the plane and 0 < λ < 1 be a real number. Given a pair (P; λ), we define the outer billiards with contraction map T as follows. For a generic point x R2 P , we can find a unique vertex v of P such that P lies on the left of the2 rayn starting from x and passing through v. Then on this ray, we pick a point y that lies on the other side of x with respect to v and satisfies xv : vy = 1 : λ. We define T x = y (Figure 1). The map T is j j j j well-defined for all points on R2 P except for points on the union of singular rays extending the sides of P .
    [Show full text]
  • Fundamental Theorems in Mathematics
    SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635].
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Dynamische Systeme
    Mathematisches Forschungsinstitut Oberwolfach Report No. 32/2017 DOI: 10.4171/OWR/2017/32 Dynamische Systeme Organised by Hakan Eliasson, Paris Helmut Hofer, Princeton Vadim Kaloshin, College Park Jean-Christophe Yoccoz, Paris 9 July – 15 July 2017 Abstract. This workshop continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser-Zehnder meeting” in 1981. The main themes of the workshop are the new results and developments in the area of dynamical systems, in particular in Hamiltonian systems and symplectic geometry. This year special emphasis where laid on symplectic methods with applications to dynamics. The workshop was dedicated to the memory of John Mather, Jean-Christophe Yoccoz and Krzysztof Wysocki. Mathematics Subject Classification (2010): 37, 53D, 70F, 70H. Introduction by the Organisers The workshop was organized by H. Eliasson (Paris), H. Hofer (Princeton) and V. Kaloshin (Maryland). It was attended by more than 50 participants from 13 countries and displayed a good mixture of young, mid-career and senior people. The workshop covered a large area of dynamical systems centered around classi- cal Hamiltonian dynamics and symplectic methods: closing lemma; Hamiltonian PDE’s; Reeb dynamics and contact structures; KAM-theory and diffusion; celes- tial mechanics. Also other parts of dynamics were represented. K. Irie presented a smooth closing lemma for Hamiltonian diffeomorphisms on closed surfaces. This result is the peak of a fantastic development in symplectic methods where, in particular, the contributions of M. Hutchings play an important role. 1988 Oberwolfach Report 32/2017 D. Peralta-Salas presented new solutions for the 3-dimensional Navier-Stokes equations with different vortex structures.
    [Show full text]
  • PENTAGONAL DOMAIN EXCHANGE Adler-Kitchens-Tresser [1] And
    PENTAGONAL DOMAIN EXCHANGE SHIGEKI AKIYAMA AND EDMUND HARRISS Abstract. Self-inducing structure of pentagonal piecewise isometry is applied to show detailed description of periodic and aperiodic orbits, and further dynamical properties. A Pisot number appears as a scaling constant and plays a crucial role in the proof. Further generalization is discussed in the last section. Adler-Kitchens-Tresser [1] and Goetz [10] initiated the study of piecewise isometries. This class of maps shows the way to possible generalizations of results on interval exchanges to higher dimensions [16, 30]. In this paper we examine the detailed properties of the map shown in Figure 1 from an algebraic point of view. Figure 1. A piecewise rotation T on two pieces. The tri- angle is rotated 2π=5 around a and the trapezium is ro- tated 2π=5 around b. Periodic points with short periods are shown below, in two colours to illustrate that they cluster into groups, each forming a pentagon. The goal of this paper is to see how this map is applied to show number theoretical results. First we reprove that almost all orbits in the sense of The first author is supported by the Japanese Society for the Promotion of Science (JSPS), grant in aid 21540010. 1 2 SHIGEKI AKIYAMA AND EDMUND HARRISS Lebesgue measure are periodic, and in addition, there are explicit aperiodic points. Second we show that aperiodic points forms a proper dense subset of an attractor of some iterated function system and are recognized by a B¨uchi automaton (c.f. Figure 14). The dynamics acting on this set of aperiodic points are conjugate to the 2-adic odometer (addition of one) whose explicit construction is given (Theorem 3).
    [Show full text]
  • 5 | Newton's Laws of Motion 207 5 | NEWTON's LAWS of MOTION
    Chapter 5 | Newton's Laws of Motion 207 5 | NEWTON'S LAWS OF MOTION Figure 5.1 The Golden Gate Bridge, one of the greatest works of modern engineering, was the longest suspension bridge in the world in the year it opened, 1937. It is still among the 10 longest suspension bridges as of this writing. In designing and building a bridge, what physics must we consider? What forces act on the bridge? What forces keep the bridge from falling? How do the towers, cables, and ground interact to maintain stability? Chapter Outline 5.1 Forces 5.2 Newton's First Law 5.3 Newton's Second Law 5.4 Mass and Weight 5.5 Newton’s Third Law 5.6 Common Forces 5.7 Drawing Free-Body Diagrams Introduction When you drive across a bridge, you expect it to remain stable. You also expect to speed up or slow your car in response to traffic changes. In both cases, you deal with forces. The forces on the bridge are in equilibrium, so it stays in place. In contrast, the force produced by your car engine causes a change in motion. Isaac Newton discovered the laws of motion that describe these situations. Forces affect every moment of your life. Your body is held to Earth by force and held together by the forces of charged particles. When you open a door, walk down a street, lift your fork, or touch a baby’s face, you are applying forces. Zooming in deeper, your body’s atoms are held together by electrical forces, and the core of the atom, called the nucleus, is held together by the strongest force we know—strong nuclear force.
    [Show full text]