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Classical Mechanics Stony Brook University Academic Commons Essential Graduate Physics Department of Physics and Astronomy 2013 Part CM: Classical Mechanics Konstantin Likharev SUNY Stony Brook, [email protected] Follow this and additional works at: https://commons.library.stonybrook.edu/egp Part of the Physics Commons Recommended Citation Likharev, Konstantin, "Part CM: Classical Mechanics" (2013). Essential Graduate Physics. 2. https://commons.library.stonybrook.edu/egp/2 This Book is brought to you for free and open access by the Department of Physics and Astronomy at Academic Commons. It has been accepted for inclusion in Essential Graduate Physics by an authorized administrator of Academic Commons. For more information, please contact [email protected], [email protected]. Konstantin K. Likharev Essential Graduate Physics Lecture Notes and Problems Beta version Open online access at http://commons.library.stonybrook.edu/egp/ and https://sites.google.com/site/likharevegp/ Part CM: Classical Mechanics Last corrections: September 10, 2021 A version of this material was published in 2017 under the title Classical Mechanics: Lecture notes IOPP, Essential Advanced Physics – Volume 1, ISBN 978-0-7503-1398-8, with model solutions of the exercise problems published in 2018 under the title Classical Mechanics: Problems with solutions IOPP, Essential Advanced Physics – Volume 2, ISBN 978-0-7503-1401-5 However, by now this online version of the lecture notes and the problem solutions available from the author, have been better corrected Note also: Konstantin K. Likharev (ed.) Essential Quotes for Scientists and Engineers Springer, 2021, ISBN 978-3-030-63331-8 (see https://essentialquotes.wordpress.com/) © K. Likharev Essential Graduate Physics CM: Classical Mechanics Table of Contents Chapter 1. Review of Fundamentals (14 pp.) 1.0. Terminology: Mechanics and dynamics 1.1. Kinematics: Basic notions 1.2. Dynamics: Newton laws 1.3. Conservation laws 1.4. Potential energy and equilibrium 1.5. OK, can we go home now? 1.6. Self-test problems (12) Chapter 2. Lagrangian Analytical Mechanics (14 pp.) 2.1. Lagrange equation 2.2. Three simple examples 2.3. Hamiltonian function and energy 2.4. Other conservation laws 2.5. Exercise problems (11) Chapter 3. A Few Simple Problems (20 pp.) 3.1. One-dimensional and 1D-reducible systems 3.2. Equilibrium and stability 3.3. Hamiltonian 1D systems 3.4. Planetary problems 3.5. Elastic scattering 3.6. Exercise problems (18) Chapter 4. Rigid Body Motion (30 pp.) 4.1. Translation and rotation 4.2. Inertia tensor 4.3. Fixed-axis rotation 4.4. Free rotation 4.5. Torque-induced precession 4.6. Non-inertial reference frames 4.7. Exercise problems (23) Chapter 5. Oscillations (34 pp.) 5.1. Free and forced oscillations 5.2. Weakly nonlinear oscillations 5.3. Reduced equations 5.4. Self-oscillations and phase locking 5.5. Parametric excitation 5.6. Fixed point classification 5.7. Numerical approaches 5.8. Harmonic and subharmonic oscillations 5.9. Exercise problems (16) Table of Contents Page 2 of 4 Essential Graduate Physics CM: Classical Mechanics Chapter 6. From Oscillations to Waves (28 pp.) 6.1. Two coupled oscillators 6.2. N coupled oscillators 6.3. 1D waves 6.4. Acoustic waves 6.5. Standing waves 6.6. Wave decay and attenuation 6.7. Nonlinear and parametric effects 6.8. Exercise problems (20) Chapter 7. Deformations and Elasticity (38 pp.) 7.1. Strain 7.2. Stress 7.3. Hooke’s law 7.4. Equilibrium 7.5. Rod bending 7.6. Rod torsion 7.7. 3D acoustic waves 7.8. Elastic waves in thin rods 7.9. Exercise problems (17) Chapter 8. Fluid Mechanics (28 pp.) 8.1. Hydrostatics 8.2. Surface tension effects 8.3. Kinematics 8.4. Dynamics: Ideal fluids 8.5. Dynamics: Viscous fluids 8.6. Turbulence 8.7. Exercise problems (19) Chapter 9. Deterministic Chaos (14 pp.) 9.1. Chaos in maps 9.2. Chaos in dynamic systems 9.3. Chaos in Hamiltonian systems 9.4. Chaos and turbulence 9.5. Exercise problems (4) Chapter 10. A Bit More of Analytical Mechanics (16 pp.) 10.1. Hamilton equations 10.2. Adiabatic invariance 10.3. The Hamilton principle 10.4. The Hamilton-Jacobi equation 10.5. Exercise problems (9) * * * Additional file (available from the author upon request): Exercise and Test Problems with Model Solutions (149 + 34 = 183 problems; 240 pp.) Table of Contents Page 3 of 4 Essential Graduate Physics CM: Classical Mechanics This page is intentionally left blank Table of Contents Page 4 of 4 Essential Graduate Physics CM: Classical Mechanics Chapter 1. Review of Fundamentals After elaborating a bit on the title and contents of the course, this short introductory chapter reviews the basic notions and facts of the non-relativistic classical mechanics, that are supposed to be known to the readers from undergraduate studies.1 Due to this reason, the discussion is very brief. 1.0. Terminology: Mechanics and dynamics A more fair title of this course would be Classical Mechanics and Dynamics, because the notions of mechanics and dynamics, though much intertwined, are still somewhat different. The term mechanics, in its narrow sense, means deriving the equations of motion of point-like particles and their systems (including solids and fluids), solution of these equations, and interpretation of the results. Dynamics is a more ambiguous term; it may mean, in particular: (i) the part of physics that deals with motion (in contrast to statics); (ii) the part of physics that deals with reasons for motion (in contrast to kinematics); (iii) the part of mechanics that focuses on its two last tasks, i.e. the solution of the equations of motion and discussion of the results.2 Because of this ambiguity, after some hesitation, I have opted to use the traditional name Classical Mechanics, implying its broader meaning that includes (similarly to Quantum Mechanics and Statistical Mechanics) studies of dynamics of some non-mechanical systems as well. 1.1. Kinematics: Basic notions The basic notions of kinematics may be defined in various ways, and some mathematicians pay much attention to analyzing such systems of axioms, and relations between them. In physics, we typically stick to less rigorous ways (in order to proceed faster to particular problems) and end debating any definition as soon as “everybody in the room” agrees that we are all speaking about the same thing – 1 The reader is advised to perform (perhaps after reading this chapter as a reminder) a self-check by solving a few problems of the dozen listed in Sec. 1.6. If the results are not satisfactory, it may make sense to start from some remedial reading. For that, I could recommend, for example (in the alphabetical order): - G. Fowles and G. Cassiday, Analytical Mechanics, 7th ed., Brooks Cole, 2004; - J. Marion and S. Thornton, Classical Dynamics of Particles and Systems, 5th ed., Saunders, 2003; - K. Symon, Mechanics, 3rd ed., Addison-Wesley, 1971. 2 The reader may have noticed that the last definition of dynamics is suspiciously close to the part of mathematics devoted to the differential equation analysis; what is the difference? An important bit of philosophy: physics may be defined as an art (and a bit of science :-) of describing Mother Nature by mathematical means; hence in many cases the approaches of a mathematician and a physicist to a problem are very similar. The main difference between them is that physicists try to express the results of their analyses in terms of properties of the systems under study, rather than the functions describing them, and as a result develop a sort of intuition (“gut feeling”) about how other similar systems may behave, even if their exact equations of motion are somewhat different – or not known at all. The intuition so developed has an enormous heuristic power, and most discoveries in physics have been made through gut-feeling-based insights rather than by just plugging one formula into another one. © K. Likharev Essential Graduate Physics CM: Classical Mechanics at least in the context they are discussed. Let me hope that the following notions used in classical mechanics do satisfy this criterion in our “room”: (i) All the Euclidean geometry notions, including the point, the straight line, the plane, etc. (ii) Reference frames: platforms for observation and mathematical description of physical phenomena. A reference frame includes a coordinate system used for measuring the point’s position (namely, its radius vector r that connects the coordinate origin to the point – see Fig. 1) and a clock that measures time t. A coordinate system may be understood as a certain method of expressing the radius vector r of a point as a set of its scalar coordinates. The most important of such systems (but by no 3 means the only one) are the Cartesian (orthogonal, linear) coordinates rj of a point, in which its radius vector may be represented as the following sum: 3 r n r , (1.1) Cartesian j j coordinates j1 4 where n1, n2, and n3 are unit vectors directed along the coordinate axis – see Fig. 1. r3 z n3 point r Fig. 1.1. Cartesian coordinates of a point. 0 n 2 r2 y n1 r1 x (iii) The absolute (“Newtonian”) space/time,5 which does not depend on the matter distribution. The space is assumed to have the Euclidean metric, which may be expressed as the following relation between the length r of any radius vector r and its Cartesian coordinates: 1/ 2 3 r r r 2 , (1.2) Euclidean j metric j1 while time t is assumed to runs similarly in all reference frames. (iv) The (instant) velocity of the point, 3 In this series, the Cartesian coordinates (introduced in 1637 by René Descartes, a.k.a.
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