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AN INTRODUCTION to LAGRANGIAN MECHANICS Alain
AN INTRODUCTION TO LAGRANGIAN MECHANICS Alain J. Brizard Department of Chemistry and Physics Saint Michael’s College, Colchester, VT 05439 July 7, 2007 i Preface The original purpose of the present lecture notes on Classical Mechanics was to sup- plement the standard undergraduate textbooks (such as Marion and Thorton’s Classical Dynamics of Particles and Systems) normally used for an intermediate course in Classi- cal Mechanics by inserting a more general and rigorous introduction to Lagrangian and Hamiltonian methods suitable for undergraduate physics students at sophomore and ju- nior levels. The outcome of this effort is that the lecture notes are now meant to provide a self-consistent introduction to Classical Mechanics without the need of any additional material. It is expected that students taking this course will have had a one-year calculus-based introductory physics course followed by a one-semester course in Modern Physics. Ideally, students should have completed their three-semester calculus sequence by the time they enroll in this course and, perhaps, have taken a course in ordinary differential equations. On the other hand, this course should be taken before a rigorous course in Quantum Mechanics in order to provide students with a sound historical perspective involving the connection between Classical Physics and Quantum Physics. Hence, the second semester of the sophomore year or the fall semester of the junior year provide a perfect niche for this course. The structure of the lecture notes presented here is based on achieving several goals. As a first goal, I originally wanted to model these notes after the wonderful monograph of Landau and Lifschitz on Mechanics, which is often thought to be too concise for most undergraduate students. -
Chapter 8 Relationship Between Rotational Speed and Magnetic
Chapter 8 Relationship Between Rotational Speed and Magnetic Fields 8.1 Introduction It is well-known that sunspots and other solar magnetic features rotate faster than the surface plasma (Howard & Harvey, 1970). The sidereal rotation speeds of weak magnetic features and plages (Howard, Gilman, & Gilman, 1984; Komm, Howard, & Harvey, 1993), individual sunspots (Howard, Gilman, & Gilman, 1984; Sivaraman, Gupta, & Howard, 1993) and sunspot groups (Howard, 1992) have been studied for many years by different research groups by use of different observatory data (see reviews of Howard 1996 and Beck 2000). Such studies are deemed as diagnostics of subsurface conditions based on the assumption that the faster rotation of magnetic features was caused by the faster-rotating plasma in the interior where the magnetic features were anchored (Gilman & Foukal, 1979). By matching the sunspot surface speed with the solar interior rotational speed robustly derived by helioseismology (e.g., Thompson et al., 1996; Kosovichev et al., 1997; Howe et al., 2000a), several studies estimated the depth of sunspot roots for sunspots of different sizes and life spans (e.g., Hiremath, 2002; Sivaraman et al., 2003). Such studies are valuable for understanding the solar dynamo and origin of solar active regions. However, D’Silva & Howard (1994) proposed a different explanation, namely that effects of buoyancy 111 112 CHAPTER 8. ROTATIONAL SPEED AND MAGNETIC FIELDS and drag coupled with the Coriolis force during the emergence of active regions might lead to the faster rotational speed. Although many studies were done to infer the rotational speed of various solar magnetic features, the relationship between the rotational speed of the magnetic fea- tures and their magnetic strength has not yet been studied. -
Rotating Objects Tend to Keep Rotating While Non- Rotating Objects Tend to Remain Non-Rotating
12 Rotational Motion Rotating objects tend to keep rotating while non- rotating objects tend to remain non-rotating. 12 Rotational Motion In the absence of an external force, the momentum of an object remains unchanged— conservation of momentum. In this chapter we extend the law of momentum conservation to rotation. 12 Rotational Motion 12.1 Rotational Inertia The greater the rotational inertia, the more difficult it is to change the rotational speed of an object. 1 12 Rotational Motion 12.1 Rotational Inertia Newton’s first law, the law of inertia, applies to rotating objects. • An object rotating about an internal axis tends to keep rotating about that axis. • Rotating objects tend to keep rotating, while non- rotating objects tend to remain non-rotating. • The resistance of an object to changes in its rotational motion is called rotational inertia (sometimes moment of inertia). 12 Rotational Motion 12.1 Rotational Inertia Just as it takes a force to change the linear state of motion of an object, a torque is required to change the rotational state of motion of an object. In the absence of a net torque, a rotating object keeps rotating, while a non-rotating object stays non-rotating. 12 Rotational Motion 12.1 Rotational Inertia Rotational Inertia and Mass Like inertia in the linear sense, rotational inertia depends on mass, but unlike inertia, rotational inertia depends on the distribution of the mass. The greater the distance between an object’s mass concentration and the axis of rotation, the greater the rotational inertia. 2 12 Rotational Motion 12.1 Rotational Inertia Rotational inertia depends on the distance of mass from the axis of rotation. -
Branched Hamiltonians and Supersymmetry
Branched Hamiltonians and Supersymmetry Thomas Curtright, University of Miami Wigner 111 seminar, 12 November 2013 Some examples of branched Hamiltonians are explored, as recently advo- cated by Shapere and Wilczek. These are actually cases of switchback poten- tials, albeit in momentum space, as previously analyzed for quasi-Hamiltonian dynamical systems in a classical context. A basic model, with a pair of Hamiltonian branches related by supersymmetry, is considered as an inter- esting illustration, and as stimulation. “It is quite possible ... we may discover that in nature the relation of past and future is so intimate ... that no simple representation of a present may exist.” – R P Feynman Based on work with Cosmas Zachos, Argonne National Laboratory Introduction to the problem In quantum mechanics H = p2 + V (x) (1) is neither more nor less difficult than H = x2 + V (p) (2) by reason of x, p duality, i.e. the Fourier transform: ψ (x) φ (p) ⎫ ⎧ x ⎪ ⎪ +i∂/∂p ⎪ ⇐⇒ ⎪ ⎬⎪ ⎨⎪ i∂/∂x p − ⎪ ⎪ ⎪ ⎪ ⎭⎪ ⎩⎪ This equivalence of (1) and (2) is manifest in the QMPS formalism, as initiated by Wigner (1932), 1 2ipy/ f (x, p)= dy x + y ρ x y e− π | | − 1 = dk p + k ρ p k e2ixk/ π | | − where x and p are on an equal footing, and where even more general H (x, p) can be considered. See CZ to follow, and other talks at this conference. Or even better, in addition to the excellent books cited at the conclusion of Professor Schleich’s talk yesterday morning, please see our new book on the subject ... Even in classical Hamiltonian mechanics, (1) and (2) are equivalent under a classical canonical transformation on phase space: (x, p) (p, x) ⇐⇒ − But upon transitioning to Lagrangian mechanics, the equivalence between the two theories becomes obscure. -
1 the Basic Set-Up 2 Poisson Brackets
MATHEMATICS 7302 (Analytical Dynamics) YEAR 2016–2017, TERM 2 HANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS These notes are intended to be read as a supplement to the handout from Gregory, Classical Mechanics, Chapter 14. 1 The basic set-up I assume that you have already studied Gregory, Sections 14.1–14.4. The following is intended only as a succinct summary. We are considering a system whose equations of motion are written in Hamiltonian form. This means that: 1. The phase space of the system is parametrized by canonical coordinates q =(q1,...,qn) and p =(p1,...,pn). 2. We are given a Hamiltonian function H(q, p, t). 3. The dynamics of the system is given by Hamilton’s equations of motion ∂H q˙i = (1a) ∂pi ∂H p˙i = − (1b) ∂qi for i =1,...,n. In these notes we will consider some deeper aspects of Hamiltonian dynamics. 2 Poisson brackets Let us start by considering an arbitrary function f(q, p, t). Then its time evolution is given by n df ∂f ∂f ∂f = q˙ + p˙ + (2a) dt ∂q i ∂p i ∂t i=1 i i X n ∂f ∂H ∂f ∂H ∂f = − + (2b) ∂q ∂p ∂p ∂q ∂t i=1 i i i i X 1 where the first equality used the definition of total time derivative together with the chain rule, and the second equality used Hamilton’s equations of motion. The formula (2b) suggests that we make a more general definition. Let f(q, p, t) and g(q, p, t) be any two functions; we then define their Poisson bracket {f,g} to be n def ∂f ∂g ∂f ∂g {f,g} = − . -
Fermat's Principle and the Geometric Mechanics of Ray
Fermat’s Principle and the Geometric Mechanics of Ray Optics Summer School Lectures, Fields Institute, Toronto, July 2012 Darryl D Holm Imperial College London [email protected] http://www.ma.ic.ac.uk/~dholm/ Texts for the course include: Geometric Mechanics I: Dynamics and Symmetry, & II: Rotating, Translating and Rolling, by DD Holm, World Scientific: Imperial College Press, Singapore, Second edition (2011). ISBN 978-1-84816-195-5 and ISBN 978-1-84816-155-9. Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, by DD Holm,T Schmah and C Stoica. Oxford University Press, (2009). ISBN 978-0-19-921290-3 Introduction to Mechanics and Symmetry, by J. E. Marsden and T. S. Ratiu Texts in Applied Mathematics, Vol. 75. New York: Springer-Verlag (1994). 1 GeometricMechanicsofFermatRayOptics DDHolm FieldsInstitute,Toronto,July2012 2 Contents 1 Mathematical setting 5 2 Fermat’s principle 8 2.1 Three-dimensional eikonal equation . 10 2.2 Three-dimensional Huygens wave fronts . 17 2.3 Eikonal equation for axial ray optics . 23 2.4 The eikonal equation for mirages . 29 2.5 Paraxial optics and classical mechanics . 32 3 Lecture 2: Hamiltonian formulation of axial ray optics 34 3.1 Geometry, phase space and the ray path . 36 3.2 Legendre transformation . 39 4 Hamiltonian form of optical transmission 42 4.1 Translation-invariant media . 49 4.2 Axisymmetric, translation-invariant materials . 50 4.3 Hamiltonian optics in polar coordinates . 53 4.4 Geometric phase for Fermat’s principle . 56 4.5 Skewness . 58 4.6 Lagrange invariant: Poisson bracket relations . 63 GeometricMechanicsofFermatRayOptics DDHolm FieldsInstitute,Toronto,July2012 3 5 Axisymmetric invariant coordinates 69 6 Geometry of invariant coordinates 73 6.1 Flows of Hamiltonian vector fields . -
Leonhard Euler: His Life, the Man, and His Works∗
SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St. -
Integrals of Motion Key Point #15 from Bachelor Course
KeyKey12-10-07 see PointPoint http://www.strw.leidenuniv.nl/˜ #15#10 fromfrom franx/college/ BachelorBachelor mf-sts-07-c4-5 Course:Course:12-10-07 see http://www.strw.leidenuniv.nl/˜ IntegralsIntegrals ofof franx/college/ MotionMotion mf-sts-07-c4-6 4.2 Constants and Integrals of motion Integrals (BT 3.1 p 110-113) Much harder to define. E.g.: 1 2 Energy (all static potentials): E(⃗x,⃗v)= 2 v + Φ First, we define the 6 dimensional “phase space” coor- dinates (⃗x,⃗v). They are conveniently used to describe Lz (axisymmetric potentials) the motions of stars. Now we introduce: L⃗ (spherical potentials) • Integrals constrain geometry of orbits. • Constant of motion: a function C(⃗x,⃗v, t) which is constant along any orbit: examples: C(⃗x(t1),⃗v(t1),t1)=C(⃗x(t2),⃗v(t2),t2) • 1. Spherical potentials: E,L ,L ,L are integrals of motion, but also E,|L| C is a function of ⃗x, ⃗v, and time t. x y z and the direction of L⃗ (given by the unit vector ⃗n, Integral of motion which is defined by two independent numbers). ⃗n de- • : a function I(x, v) which is fines the plane in which ⃗x and ⃗v must lie. Define coor- constant along any orbit: dinate system with z axis along ⃗n I[⃗x(t1),⃗v(t1)] = I[⃗x(t2),⃗v(t2)] ⃗x =(x1,x2, 0) I is not a function of time ! Thus: integrals of motion are constants of motion, ⃗v =(v1,v2, 0) but constants of motion are not always integrals of motion! → ⃗x and ⃗v constrained to 4D region of the 6D phase space. -
Schrödinger's Argument.Pdf
SCHRÖDINGER’S TRAIN OF THOUGHT Nicholas Wheeler, Reed College Physics Department April 2006 Introduction. David Griffiths, in his Introduction to Quantum Mechanics (2nd edition, 2005), is content—at equation (1.1) on page 1—to pull (a typical instance of) the Schr¨odinger equation 2 2 i ∂Ψ = − ∂ Ψ + V Ψ ∂t 2m ∂x2 out of his hat, and then to proceed directly to book-length discussion of its interpretation and illustrative physical ramifications. I remarked when I wrote that equation on the blackboard for the first time that in a course of my own design I would feel an obligation to try to encapsulate the train of thought that led Schr¨odinger to his equation (1926), but that I was determined on this occasion to adhere rigorously to the text. Later, however, I was approached by several students who asked if I would consider interpolating an account of the historical events I had felt constrained to omit. That I attempt to do here. It seems to me a story from which useful lessons can still be drawn. 1. Prior events. Schr¨odinger cultivated soil that had been prepared by others. Planck was led to write E = hν by his successful attempt (1900) to use the then-recently-established principles of statistical mechanics to account for the spectral distribution of thermalized electromagnetic radiation. It was soon appreciated that Planck’s energy/frequency relation was relevant to the understanding of optical phenomena that take place far from thermal equilibrium (photoelectric effect: Einstein 1905; atomic radiation: Bohr 1913). By 1916 it had become clear to Einstein that, while light is in some contexts well described as a Maxwellian wave, in other contexts it is more usefully thought of as a massless particle, with energy E = hν and momentum p = h/λ. -
Hamilton's Principle in Continuum Mechanics
Hamilton’s Principle in Continuum Mechanics A. Bedford University of Texas at Austin This document contains the complete text of the monograph published in 1985 by Pitman Publishing, Ltd. Copyright by A. Bedford. 1 Contents Preface 4 1 Mechanics of Systems of Particles 8 1.1 The First Problem of the Calculus of Variations . 8 1.2 Conservative Systems . 12 1.2.1 Hamilton’s principle . 12 1.2.2 Constraints.......................... 15 1.3 Nonconservative Systems . 17 2 Foundations of Continuum Mechanics 20 2.1 Mathematical Preliminaries . 20 2.1.1 Inner Product Spaces . 20 2.1.2 Linear Transformations . 22 2.1.3 Functions, Continuity, and Differentiability . 24 2.1.4 Fields and the Divergence Theorem . 25 2.2 Motion and Deformation . 27 2.3 The Comparison Motion . 32 2.4 The Fundamental Lemmas . 36 3 Mechanics of Continuous Media 39 3.1 The Classical Theories . 40 3.1.1 IdealFluids.......................... 40 3.1.2 ElasticSolids......................... 46 3.1.3 Inelastic Materials . 50 3.2 Theories with Microstructure . 54 3.2.1 Granular Solids . 54 3.2.2 Elastic Solids with Microstructure . 59 2 4 Mechanics of Mixtures 65 4.1 Motions and Comparison Motions of a Mixture . 66 4.1.1 Motions............................ 66 4.1.2 Comparison Fields . 68 4.2 Mixtures of Ideal Fluids . 71 4.2.1 Compressible Fluids . 71 4.2.2 Incompressible Fluids . 73 4.2.3 Fluids with Microinertia . 75 4.3 Mixture of an Ideal Fluid and an Elastic Solid . 83 4.4 A Theory of Mixtures with Microstructure . 86 5 Discontinuous Fields 91 5.1 Singular Surfaces . -
STUDENT HIGHLIGHT – Hassan Najafi Alishah - “Symplectic Geometry, Poisson Geometry and KAM Theory” (Mathematics)
SEPTEMBER OCTOBER // 2010 NUMBER// 28 STUDENT HIGHLIGHT – Hassan Najafi Alishah - “Symplectic Geometry, Poisson Geometry and KAM theory” (Mathematics) Hassan Najafi Alishah, a CoLab program student specializing in math- The Hamiltonian function (or energy function) is constant along the ematics, is doing advanced work in the Mathematics Departments of flow. In other words, the Hamiltonian function is a constant of motion, Instituto Superior Técnico, Lisbon, and the University of Texas at Austin. a principle that is sometimes called the energy conservation law. A This article provides an overview of his research topics. harmonic spring, a pendulum, and a one-dimensional compressible fluid provide examples of Hamiltonian Symplectic and Poisson geometries underlie all me- systems. In the former, the phase spaces are symplec- chanical systems including the solar system, the motion tic manifolds, but for the later phase space one needs of a rigid body, or the interaction of small molecules. the more general concept of a Poisson manifold. The origins of symplectic and Poisson structures go back to the works of the French mathematicians Joseph-Louis A Hamiltonian system with enough suitable constants Lagrange (1736-1813) and Simeon Denis Poisson (1781- of motion can be integrated explicitly and its orbits can 1840). Hermann Weyl (1885-1955) first used “symplectic” be described in detail. These are called completely in- with its modern mathematical meaning in his famous tegrable Hamiltonian systems. Under appropriate as- monograph “The Classical groups.” (The word “sym- sumptions (e.g., in a compact symplectic manifold) plectic” itself is derived from a Greek word that means the motions of a completely integrable Hamiltonian complex.) Lagrange introduced the concept of a system are quasi-periodic. -
Euler and Chebyshev: from the Sphere to the Plane and Backwards Athanase Papadopoulos
Euler and Chebyshev: From the sphere to the plane and backwards Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Euler and Chebyshev: From the sphere to the plane and backwards. 2016. hal-01352229 HAL Id: hal-01352229 https://hal.archives-ouvertes.fr/hal-01352229 Preprint submitted on 6 Aug 2016 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. EULER AND CHEBYSHEV: FROM THE SPHERE TO THE PLANE AND BACKWARDS ATHANASE PAPADOPOULOS Abstract. We report on the works of Euler and Chebyshev on the drawing of geographical maps. We point out relations with questions about the fitting of garments that were studied by Chebyshev. This paper will appear in the Proceedings in Cybernetics, a volume dedicated to the 70th anniversary of Academician Vladimir Betelin. Keywords: Chebyshev, Euler, surfaces, conformal mappings, cartography, fitting of garments, linkages. AMS classification: 30C20, 91D20, 01A55, 01A50, 53-03, 53-02, 53A05, 53C42, 53A25. 1. Introduction Euler and Chebyshev were both interested in almost all problems in pure and applied mathematics and in engineering, including the conception of industrial ma- chines and technological devices. In this paper, we report on the problem of drawing geographical maps on which they both worked.