MATH0054 Analytical Dynamics

Total Page：16

File Type：pdf, Size：1020Kb

MATH0054 Analytical Dynamics Year: 2021{2022 Code: MATH0054 Level: 6 (UG) Normal student group(s): UG: Year 3 Mathematics degrees Value: 15 credits (= 7.5 ECTS credits) Term: 2 Assessment: 90% examination, 10% coursework Normal Pre-requisites: MATH0009, MATH0011 Lecturer: Prof A Sokal Course Description and Objectives Analytical dynamics develops Newtonian mechanics to the stage where powerful mathematical techniques can be used to determine the behaviour of many physical systems. The mathematical framework also plays a role in the formulation of modern quantum and relativity theories. Topics studied are the kinematics of frames of reference (including rotating frames), dynamics of systems of particles, Lagrangian and Hamiltonian dynamics and rigid body dynamics. The emphasis is both on the formal development of the theory and also use of theory in solving actual physical problems. Recommended Texts Relevant books are: (i) Gregory, Classical Mechanics; (ii) T L Chow, Classical Mechanics; (iii) Goldstein, Classical Mechanics; (iv) Taylor, Classical Mechanics; (v) Marion, Classical Dynamics of Particles and Systems Detailed Syllabus − Review of the fundamental principles of Newtonian mechanics; Galileo's principle of relativity. − Systems of particles and conservation laws: linear momentum, angular momentum, energy (internal and external potentials). − Systems of coupled linear oscillators: normal modes. − Introduction to perturbation theory for anharmonic oscillators. − Lagrangian dynamics: generalised coordinates, variational principles, symmetries and conservation laws. − Hamiltonian dynamics: phase space, Poisson brackets, introduction to canonical transfor- mations. − Kinematics and dynamics in noninertial reference frames: centrifugal and Coriolis pseudo- forces. − Rigid bodies: Eulerian angles, inertia matrices, Euler's equations of motions, force-free motion, tops. May 2021 MATH0054.

Copy Link

Recommended publications

All-Master-File-Problem-Set
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 THE CITY UNIVERSITY OF NEW YORK First Examination for PhD Candidates in Physics Analytical Dynamics Summer 2010 Do two of the following three problems. Start each problem on a new page. Indicate clearly which two problems you choose to solve. If you do not indicate which problems you wish to be graded, only the ﬁrst two problems will be graded. Put your identiﬁcation number on each page. 1. A particle moves without friction on the axially symmetric surface given by 1 z = br2, x = r cos φ, y = r sin φ 2 where b > 0 is constant and z is the vertical direction. The particle is subject to a homogeneous gravitational force in z-direction given by −mg where g is the gravitational acceleration. (a) (2 points) Write down the Lagrangian for the system in terms of the generalized coordinates r and φ. (b) (3 points) Write down the equations of motion. (c) (5 points) Find the Hamiltonian of the system. (d) (5 points) Assume that the particle is moving in a circular orbit at height z = a. Obtain its energy and angular momentum in terms of a, b, g. (e) (10 points) The particle in the horizontal orbit is poked downwards slightly. Obtain the frequency of oscillation about the unperturbed orbit for a very small oscillation amplitude. 2. Use relativistic dynamics to solve the following problem: A particle of rest mass m and initial velocity v0 along the x-axis is subject after t = 0 to a constant force F acting in the y-direction.
[Show full text]
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 .
[Show full text]
Introduction and Basic Concepts
ANALYTICAL MECHANICS Introduction and Basic Concepts Paweª FRITZKOWSKI, Ph.D. Eng. Division of Technical Mechanics Institute of Applied Mechanics Faculty of Mechanical Engineering and Management POZNAN UNIVERSITY OF TECHNOLOGY Agenda 1 Introduction to the Course 2 Degrees of Freedom and Constraints 3 Generalized Quantities 4 Problems 5 Summary 6 Bibliography Paweª Fritzkowski Introduction: Basic Concepts 2 / 37 1. Introduction to the Course Introduction to the Course Analytical mechanics What is it all about? Paweª Fritzkowski Introduction: Basic Concepts 4 / 37 Introduction to the Course Analytical mechanics What is it all about? Analytical mechanics... is a branch of classical mechanics results from a reformulation of the classical Galileo's and Newton's concepts is an approach dierent from the vector Newtonian mechanics: more advanced, sophisticated and mathematically-oriented eliminates the need to analyze forces on isolated parts of mechanical systems is a more global way of thinking: allows one to treat a system as a whole Paweª Fritzkowski Introduction: Basic Concepts 5 / 37 Introduction to the Course Analytical mechanics What is it all about? (cont.) provides more powerful and easier ways to derive equations of motion, even for complex mechanical systems is based on some scalar functions which describe an entire system is a common tool for creating mathematical models for numerical simulations has spread far beyond the pure mechanics and inuenced various areas of physics Paweª Fritzkowski Introduction: Basic Concepts 6 / 37
[Show full text]
On the Foundations of Analytical Dynamics F.E
International Journal of Non-Linear Mechanics 37 (2002) 1079–1090 On the foundations of analytical dynamics F.E. Udwadiaa; ∗, R.E. Kalabab aAerospace and Mechanical Engineering, Civil Engineering, Mathematics, and Information and Operations Management, 430K Olin Hall, University of Southern California, Los Angeles, CA 90089-1453, USA bBiomedical Engineering and Economics, University of Southern California, Los Angeles, CA 90089, USA Abstract In this paper, we present the general structure for the explicit equations of motion for general mechanical systems subjected to holonomic and non-holonomic equality constraints. The constraints considered here need not satisfy D’Alembert’s principle, and our derivation is not based on the principle of virtual work. Therefore, the equations obtained here have general applicability. They show that in the presence of such constraints, the constraint force acting on the systemcan always be viewed as madeup of the sumof two components.The explicit formfor each of the two components is provided. The ÿrst of these components is the constraint force that would have existed, were all the constraints ideal; the second is caused by the non-ideal nature of the constraints, and though it needs speciÿcation by the mechanician and depends on the particular situation at hand, this component nonetheless has a speciÿc form. The paper also provides a generalized formof D’Alembert’sprinciple which is then used to obtain the explicit equations of motion for constrained mechanical systems where the constraints may be non-ideal. We show an example where the new general, explicit equations of motion obtained in this paper are used to directly write the equations of motion for describing a non-holonomically constrained system with non-ideal constraints.
[Show full text]
Ed Reiss Library Collection
Ed Reiss Library Collection This collection can be found in the Main Office of the Department of Engineering Sciences and Applied Mathematics in Tech M426. BOOKS Author(s) Title Year of Publication A Abramowitz M., Stegun I., Editors Handbook of Mathematical Functions 1965 Abramsom H.N. An Introduction to the Dynamics of Airplanes 1958 Achenbach J.D. Reciprocity in Elastodynamics 2003 Achenbach J.D., Pao Y.H., Tiersten Report of the Workshop on Application of Elastic Waves in Electrical Devices, Non-Destructive H.F. Testing and Seismology 1976 *Agnew, R. P. Calculus: Analytic Geometry and Calculus, with Vectors 1962 Akhiezer N.I. The Calculus of Variations 1962 Albers V.M. Underwater Sound 1972 Albers V.M. Underwater Acoustics Vol II 1967 Alfrey Jr. T. Mechanical Behavior of High Polymers 1948 Allen D.N. de G. Relaxation Methods 1954 Andrews D. An Introduction to Atmospheric Physics (2nd Edition) 2010 Apostol T.M. Mathematical Analysis: A Modern Approach to Advanced Calculus 1957 Arbocz J, Potier-Ferry M, Singer J., Tvergaard V. Lecture Notes in Physics 1985 *Arkin H., Colton R.R. An Outline of Statistical Methods 1939 Arnold L., Jinqiao D. Stochastics and Dynamics 2001 Ash E.A., Paige E.G.S. Rayleigh-Wave Theory and Application 1985 B Baily C., Comte-Bellot G. Turbulence 2003 Barrat A., Barthelemy M., Vespignani A. Dynamical Processes on Complex Networks 2008 Barton M. Fundamentals of Aircraft Structures 1948 Batchelor G.K., Moffatt H.K., Worster M.G. Perspectives in Fluid Dynamica 2003 Bellan P.M. Fundamentals of Plasma Physics 2006 Bellan P.M. Fundamentals of Plasma Physics 2008 Bellman R.
[Show full text]
MATH0054 (Analytical Dynamics)
MATH0054 (Analytical Dynamics) Year: 2019{2020 Code: MATH0054 Old Code: MATH7302 Level: 6 (UG) Normal student group(s): UG: Year 2 and 3 Mathematics degrees Value: 15 credits (= 7.5 ECTS credits) Term: 2 Structure: 3 hour lectures per week. Assessed coursework. Assessment: 90% examination, 10% coursework. In order to pass the module you must have at least 40% for both the examination mark and the ﬁnal weighted mark. Normal Pre-requisites: MATH0009 (previously MATH1302), MATH0011 (previously MATH1402) Lecturer: Prof A Sokal Course Description and Objectives Analytical dynamics develops Newtonian mechanics to the stage where powerful mathematical techniques can be used to determine the behaviour of many physical systems. The mathematical framework also plays a role in the formulation of modern quantum and relativity theories. Topics studied are the kinematics of frames of reference (including rotating frames), dynamics of systems of particles, Lagrangian and Hamiltonian dynamics and rigid body dynamics. The emphasis is both on the formal development of the theory and also use of theory in solving actual physical problems. Recommended Texts Relevant books are: (i) Gregory, Classical Mechanics; (ii) T L Chow, Classical Mechanics; (iii) Goldstein, Classical Mechanics; (iv) Taylor, Classical Mechanics; (v) Marion, Classical Dynamics of Particles and Systems Detailed Syllabus − Review of the fundamental principles of Newtonian mechanics; Galileo's principle of rel- ativity. − Systems of particles and conservation laws: linear momentum, angular momentum, energy (internal and external potentials). − Systems of coupled linear oscillators: normal modes. − Introduction to perturbation theory for anharmonic oscillators. − Lagrangian dynamics: generalised coordinates, variational principles, symmetries and conservation laws. − Hamiltonian dynamics: phase space, Poisson brackets, introduction to canonical transfor- mations.
[Show full text]
Classical Mechanics January 2012
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 THE CITY UNIVERSITY OF NEW YORK First Examination for PhD Candidates in Physics Analytical Dynamics Summer 2010 Do two of the following three problems. Start each problem on a new page. Indicate clearly which two problems you choose to solve. If you do not indicate which problems you wish to be graded, only the ﬁrst two problems will be graded. Put your identiﬁcation number on each page. 1. A particle moves without friction on the axially symmetric surface given by 1 z = br2, x = r cos φ, y = r sin φ 2 where b > 0 is constant and z is the vertical direction. The particle is subject to a homogeneous gravitational force in z-direction given by −mg where g is the gravitational acceleration. (a) (2 points) Write down the Lagrangian for the system in terms of the generalized coordinates r and φ. (b) (3 points) Write down the equations of motion. (c) (5 points) Find the Hamiltonian of the system. (d) (5 points) Assume that the particle is moving in a circular orbit at height z = a. Obtain its energy and angular momentum in terms of a, b, g. (e) (10 points) The particle in the horizontal orbit is poked downwards slightly. Obtain the frequency of oscillation about the unperturbed orbit for a very small oscillation amplitude. 2. Use relativistic dynamics to solve the following problem: A particle of rest mass m and initial velocity v0 along the x-axis is subject after t = 0 to a constant force F acting in the y-direction.
[Show full text]
Analytical Dynamics: Lagrange's Equation and Its Application
Analytical Dynamics: Lagrange’s Equation and its Application – A Brief Introduction D. S. Stutts, Ph.D. Associate Professor of Mechanical Engineering Missouri University of Science and Technology Rolla, MO 65409-0050 [email protected] April 9, 2017 Contents 1 The Calculus of Variations 1 1.1 Extremum of an Integral – The Euler-Lagrange Equation .................. 2 2 Hamilton’s Principle 4 2.1 Generalized Coordinates and Forces .............................. 5 3 Lagrange’s Equations of Motion 9 3.1 Lagrange’s Equations Via The Extended Hamilton’s Principle ................ 9 3.2 Rayleigh’s Dissipation function ................................. 10 3.3 Kinematic Requirements of Lagrange’s Equation ....................... 10 3.4 Lagrange Equation Examples .................................. 11 4 Constrained Maxima and Minima 13 4.1 Lagrange’s Multipliers ...................................... 14 4.2 Application of Lagrange Multipliers to Compute Equilibrium Reaction Forces ....... 16 4.3 Application of Lagrange Multipliers to Compute Dynamic Reaction Forces ......... 18 4.3.1 Gear Train Constraint Force Example ......................... 18 5 Hamilton’s Principle: Continuous Models 20 6 Bibliography 23 Index 23 1 The Calculus of Variations The Euler-Lagrange formulation was built upon the foundation of the the calculus of variations, the initial development of which is usually credited to Leonhard Euler.1 The calculus of variations is an extensive subject, and there are many fine references which present a detailed development of the subject– see 1http://en.wikipedia.org/wiki/Calculus_of_variations 1 ©Dr. D. S. Stutts, 1995 – 2017. 1 THE CALCULUS OF VARIATIONS Bibliography. The purpose of this addendum is do provide a brief background in the theory behind La- grange’s Equations. Fortunately, complete understanding of this theory is not absolutely necessary to use Lagrange’s equations, but a basic understanding of variational principles can greatly increase your mechanical modeling skills.
[Show full text]
Analytical Dynamics: Lagrange's Equation and Its Application
Analytical Dynamics: Lagrange's Equation and its Application { A Brief Introduction D. S. Stutts, Ph.D. Associate Professor of Mechanical Engineering Missouri University of Science and Technology Rolla, MO 65409-0050 January 24, 2011∗ ∗Corrected typo in Equation (6) { thanks, Shantanu! 1 © Dr. Daniel S. Stutts, 1995 { 2011All Rights Res. 1 THE CALCULUS OF VARIATIONS 1 The Calculus of Variations The calculus of variations is an extensive subject, and there are many ﬁne references which present a detailed development of the subject { see Bibliography. The purpose of this addendum is do provide a brief background in the theory behind Lagrange's Equations. Fortunately, complete understanding of this theory is not absolutely necessary to use Lagrange's equations, but a basic understanding of variational principles can greatly increase your mechanical modeling skills. 1.1 Extremum of an Integral { The Euler-Lagrange Equation Given the Integral of a functional (a function of functions) of the form Z t2 I() = F (U; U;_ t)dt; (1) t1 _ where t1, and t2 are arbitrary, is a small positive, real constant, and U and U are given by U(t) = u(t) + η(t); and U_ (t) =u _(t) + η_(t): (2) The functions U, and u may be thought of as describing the possible positions of a dynamical system between the two instants in time, t1, and t2, where u(t) represents the position when the integral de- scribed by Equation (1) is stationary, i.e. where it is an extremum, and U(t) is u(t) plus a variation η(t).
[Show full text]
Mechanical Engineering Courses MECH 5301 Mathematical Methods
Mechanical Engineering Courses MECH 5301 Mathematical Methods for Mechanical Engineers (3-0) The primary objective of this class is to give grounding in some basic mathematical analysis methods that are most relevant to mechanical engineers. Topics include linear algebra and vector spaces, tensorial calculus, ordinary and partial differential equations, MECH 5302 Solid Mechanics I (3-0) An introduction to continuum mechanics, elasticity, stress, strain and constitutive models for solid mechanics problems. MECH 5303 Heat Transfer I (3-0) Introduction to heat conduction and convection; steady state and transient solutions, analytical and numerical methods. MECH 5304 Heat Transfer II (3-0) Advanced topics in heat conduction, convection and radiation; Turbulent effects free convection, boundary layers. MECH 5305 Computational Fluid Mechanics (3-0) Flow models and governing equations, mathematical behavior of partial differential equation, discretization technique: finite difference and finite volume, basics of numerics: algorithms for solving systems of linear algebraic equations, numerical stability, heat conduction, convection and diffusion, calculation of the flowfield: SIMPLE and SIMPLER algorithm, grids and transformation, post processing. MECH 5306 Fluid Mechanics (3-0) Survey of the principal concepts of fluid mechanics, statics, continuity, momentum and energy relations for continuum fluids, kinematics of fluid motion, governing equations for motion of non- viscous fluid, vorticity and circulation, and Kelvin's theorem. Helmholtz theorem, Crocco's theorem, steam function, potential flow, conformal transformation, theory or lift, and wave phenomena in fluids. MECH 5310 Thermodynamics (3-0) Applications of general thermodynamic relations; study and applications of time-dependent energy relationships; analysis of power, refrigeration, cryogenic and direct energy conversion systems. MECH 5311 Nonlinear Finite Element Analysis (3-0) This class will give an introduction to theory and application of advanced nonlinear finite element analysis.
[Show full text]
An Introduction to the Classical Three-Body Problem from Periodic Solutions to Instabilities and Chaos
GENERAL ARTICLE An Introduction to the Classical Three-Body Problem From Periodic Solutions to Instabilities and Chaos Govind S Krishnaswami and Himalaya Senapati The classical three-body problem arose in an attempt to un- derstand the eﬀect of the Sun on the Moon’s Keplerian orbit around the Earth. It has attracted the attention of some of the best physicists and mathematicians and led to the discovery of ‘chaos’. We survey the three-body problem in its historical context and use it to introduce several ideas and techniques that have been developed to understand classical mechanical (Left) Govind Krishnaswami systems. is on the faculty of the Chennai Mathematical Institute. His research concerns various problems in 1. Introduction theoretical and mathematical physics. The three-body problem is one of the oldest problems in classical dynamics that continues to throw up surprises. It has challenged (Right) Himalaya Senapati is scientists from Newton’s time to the present. It arose in an at- a PhD student at the Chennai tempt to understand the Sun’s eﬀect on the motion of the Moon Mathematical Institute. He works on dynamical systems around the Earth. This was of much practical importance in ma- and chaos. rine navigation, where lunar tables were necessary to accurately determine longitude at sea (see Box 1). The study of the three-body problem led to the discovery of the planet Neptune (see Box 2), it explains the location and stability of the Trojan asteroids and has furthered our understanding of the stability of the solar system [1]. Quantum mechanical variants of Keywords the three-body problem are relevant to the helium atom and water Kepler problem, three-body prob- lem, celestial mechanics, classi- molecule [2].
[Show full text]
Arxiv:Physics/0410149V1 [Physics.Ed-Ph] 19 Oct 2004 Ftepsil Rjcoisi Eta Oc Oin Most Motion
Orbits in a central force field: Bounded orbits Subhankar Ray∗ Dept of Physics, Jadavpur University, Calcutta 700 032, India J. Shamanna Physics Department, Visva Bharati University, Santiniketan 731235, India (Dated: August 1, 2003) The nature of boundedness of orbits of a particle moving in a central force field is investigated. General conditions for circular orbits and their stability are discussed. In a bounded central field orbit, a particle moves clockwise or anticlockwise, depending on its angular momentum, and at the same time oscillates between a minimum and a maximum radial distance, defining an inner and an outer annulus. There are generic orbits suggested in popular texts displaying the general features of a central orbit. In this work it is demonstrated that some of these orbits, seemingly possible at the first glance, are not compatible with a central force field. For power law forces, the general nature of boundedness and geometric shape of orbits are investigated. I. INTRODUCTION B. Newtonian Synthesis The central force motion is one of the oldest and widely Almost 100 years later Newton realized that the plan- studied problems in classical mechanics. Several familiar ets go about in their nearly circular orbits around the sun force-laws in nature, e.g., Newton’s law of gravitation, under the influence of the same force that causes an ap- Coulomb’s law, van-der Waals force, Yukawa interaction, ple to fall to the ground, i.e., gravitation. Newton’s law and Hooke’s law are all examples of central forces. The of gravitation gave a theoretical basis to Kepler’s laws.
[Show full text]