On the Foundations of Analytical Dynamics F.E
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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. -
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 . -
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 -
1 Portraits Leonhard Euler Daniel Bernoulli Johann-Heinrich Lambert
Portraits Leonhard Euler Daniel Bernoulli Johann-Heinrich Lambert Compiled and translated by Oscar Sheynin Berlin, 2010 Copyright Sheynin 2010 www.sheynin.de ISBN 3-938417-01-3 1 Contents Foreword I. Nicolaus Fuss, Eulogy on Leonhard Euler, 1786. Translated from German II. M. J. A. N. Condorcet, Eulogy on Euler, 1786. Translated from French III. Daniel Bernoulli, Autobiography. Translated from Russian; Latin original received in Petersburg in 1776 IV. M. J. A. N. Condorcet, Eulogy on [Daniel] Bernoulli, 1785. In French. Translated by Daniel II Bernoulli in German, 1787. This translation considers both versions V. R. Wolf, Daniel Bernoulli from Basel, 1700 – 1782, 1860. Translated from German VI. Gleb K. Michajlov, The Life and Work of Daniel Bernoullli, 2005. Translated from German VII. Daniel Bernoulli, List of Contributions, 2002 VIII. J. H. S. Formey, Eulogy on Lambert, 1780. Translated from French IX. R. Wolf, Joh. Heinrich Lambert from Mühlhausen, 1728 – 1777, 1860. Translated from German X. J.-H. Lambert, List of Publications, 1970 XI. Oscar Sheynin, Supplement: Daniel Bernoulli’s Instructions for Meteorological Stations 2 Foreword Along with the main eulogies and biographies [i, ii, iv, v, viii, ix], I have included a recent biography of Daniel Bernoulli [vi], his autobiography [iii], for the first time translated from the Russian translation of the Latin original but regrettably incomplete, and lists of published works by Daniel Bernoulli [vii] and Lambert [x]. The first of these lists is readily available, but there are so many references to the works of these scientists in the main texts, that I had no other reasonable alternative. -
New General Principle of Mechanics and Its Application to General Nonideal Nonholonomic Systems
New General Principle of Mechanics and Its Application to General Nonideal Nonholonomic Systems Firdaus E. Udwadia1 Abstract: In this paper we develop a general minimum principle of analytical dynamics that is applicable to nonideal constraints. The new principle encompasses Gauss’s Principle of Least Constraint. We use this principle to obtain the general, explicit, equations of motion for holonomically and/or nonholonomically constrained systems with non-ideal constraints. Examples of a nonholonomically constrained system where the constraints are nonideal, and of a system with sliding friction, are presented. DOI: 10.1061/͑ASCE͒0733-9399͑2005͒131:4͑444͒ CE Database subject headings: Constraints; Equations of motion; Mechanical systems; Friction. Introduction ments. Such systems have, to date, been left outside the perview of the Lagrangian framework. As stated by Goldstein ͑1981, p. The motion of complex mechanical systems is often mathemati- 14͒ “This ͓total work done by forces of constraint equal to zero͔ cally modeled by what we call their equations of motion. Several is no longer true if sliding friction is present, and we must exclude formalisms ͓Lagrange’s equations ͑Lagrange 1787͒, Gibbs– such systems from our ͓Lagrangian͔ formulation.” And Pars Appell equations ͑Gibbs 1879, Appell 1899͒, generalized inverse ͑1979͒ in his treatise on analytical dynamics writes, “There are in equations ͑Udwadia and Kalaba 1992͔͒ have been developed for fact systems for which the principle enunciated ͓D’Alembert’s obtaining the equations of motion for such structural and me- principle͔… does not hold. But such systems will not be consid- chanical systems. Though these formalisms do not all afford the ered in this book.” Newtonian approaches are usually used to deal same ease of use in any given practical situation, they are equiva- with the problem of sliding friction ͑Goldstein 1981͒. -
On Stability Problem of a Top Rendiconti Del Seminario Matematico Della Università Di Padova, Tome 68 (1982), P
RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA V. V. RUMJANTSEV On stability problem of a top Rendiconti del Seminario Matematico della Università di Padova, tome 68 (1982), p. 119-128 <http://www.numdam.org/item?id=RSMUP_1982__68__119_0> © Rendiconti del Seminario Matematico della Università di Padova, 1982, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On Stability Problem of a Top. V. V. RUMJANTSEV (*) This paper deals with the stability of a heavy gyrostate [1] on a horizontal plane. The gyrostate is considered as a rigid body with a rotor rotating freely (without friction) about an axis invariably con- nected with the body leaning on a plane by a convex surface, i.e. the top in a broad sence of this word. For mechanician the top is a symple and principal object of study [2] attracting investigators’ attention. 1. Let $ql be the fixed coordinate system with the origin in some point of a horizontal plane and vertically up directed axis I with unit vector y; OXIX2Xa is the coordinate system rigidly connected with the body with the origin in centre of mass of gyrostate and axis ~3 coincided with one of its principal central axes of inertia. -
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. -
Recent Advances in Multi-Body Dynamics and Nonlinear Control
Recent Advances in Multi-body Dynamics and Nonlinear Control Recent Advances in Multi-body Firdaus E. Udwadia Dynamics and Nonlinear Control Aerospace and Mechanical Engineering This paper presents some recent advances in the dynamics and control of constrained Civil Engineering,, Mathematics multi-body systems. The constraints considered need not satisfy D’Alembert’s principle Systems Architecture Engineering and therefore the results are of general applicability. They show that in the presence of and Information and Operations Management constraints, the constraint force acting on the multi-body system can always be viewed as University of Southern California, Los Angeles made up of the sum of two components whose explicit form is provided. The first of these CA 90089-1453 components consists of the constraint force that would have existed were all the [email protected] constraints ideal; the second is caused by the non-ideal nature of the constraints, and though it needs specification by the mechanician who is modeling the specific system at hand, it nonetheless has a specific form. The general equations of motion obtained herein provide new insights into the simplicity with which Nature seems to operate. They are shown to provide new and exact methods for the tracking control of highly nonlinear mechanical and structural systems without recourse to the usual and approximate methods of linearization that are commonly in use. Keywords : Constrained motion, multi-body dynamics, explicit equations of motion, exact tracking control of nonlinear systems In this paper we extend these results along two directions. First, Introduction we extend D’Alembert’s Principle to include constraints that may be, in general, non-ideal so that the forces of constraint may The general problem of obtaining the equations of motion of a therefore do positive, negative, or zero work under virtual constrained discrete mechanical system is one of the central issues displacements at any given instant of time during the motion of the in multi-body dynamics. -
Explicit Equations of Motion for Constrained
Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics Firdaus Udwadia, Phailaung Phohomsiri To cite this version: Firdaus Udwadia, Phailaung Phohomsiri. Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2006, 462 (2071), pp.2097 - 2117. 10.1098/rspa.2006.1662. hal-01395968 HAL Id: hal-01395968 https://hal.archives-ouvertes.fr/hal-01395968 Submitted on 13 Nov 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. Distributed under a Creative Commons Attribution| 4.0 International License Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics 1 2 FIRDAUS E. UDWADIA AND PHAILAUNG PHOHOMSIRI 1Civil Engineering, Aerospace and Mechanical Engineering, Mathematics, and Information and Operations Management, and 2Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA We present the new, general, explicit form of the equations of motion for constrained mechanical systems applicable to systems with singular mass matrices. The systems may have holonomic and/or non holonomic constraints, which may or may not satisfy D’Alembert’s principle at each instant of time. -
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. -
Catalogue 176
C A T A L O G U E – 1 7 6 JEFF WEBER RARE BOOKS Catalogue 176 Revolutions in Science THE CURRENT catalogue continues the alphabet started with cat. #174. Lots of new books are being offered here, including books on astronomy, mathematics, and related fields. While there are many inexpensive books offered there are also a few special pieces, highlighted with the extraordinary LUBIENIECKI, this copy being entirely handcolored in a contemporary hand. Among the books are the mathematic libraries of Dr. Harold Levine of Stanford University and Father Barnabas Hughes of the Franciscan order in California. Additional material is offered from the libraries of David Lindberg and L. Pearce Williams. Normally I highlight the books being offered, but today’s bookselling world is changing rapidly. Many books are only sold on-line and thus many retailers have become abscent from city streets. If they stay in the trade, they deal on-line. I have come from a tradition of old style bookselling and hope to continue binging fine books available at reasonable prices as I have in the past. I have been blessed with being able to represent many collections over the years. No one could predict where we are all now today. What is your view of today’s book world? How can I serve you better? Let me know. www.WeberRareBooks.com On the site are more than 10,000 antiquarian books in the fields of science, medicine, Americana, classics, books on books and fore-edge paintings. The books in current catalogues are not listed on-line until mail-order clients have priority. -
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.