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"Lecture Notes on Classical Mechanics for Physics 106ab Sunil Golwala Revision Date: January 15, 2007 Introduction These notes were written during the Fall, 2004, and Winter, 2005, terms. They are indeed lecture notes – I literally lecture from these notes. They combine material from Hand and Finch (mostly), Thornton, and Goldstein, but cover the material in a diﬀerent order than any one of these texts and deviate from them widely in some places and less so in others. The reader will no doubt ask the question I asked myself many times while writing these notes: why bother? There are a large number of mechanics textbooks available all covering this very standard material, complete with worked examples and end-of-chapter problems. I can only defend myself by saying that all teachers understand their material in a slightly diﬀerent way and it is very diﬃcult to teach from someone else’s point of view – it’s like walking in shoes that are two sizes wrong. It is inevitable that every teacher will want to present some of the material in a way that diﬀers from the available texts. These notes simply put my particular presentation down on the page for your reference. These notes are not a substitute for a proper textbook; I have not provided nearly as many examples or illustrations, and have provided no exercises. They are a supplement. I suggest you skim them in parallel while reading one of the recommended texts for the course, focusing your attention on places where these notes deviate from the texts. ii Contents 1 Elementary Mechanics 1.1 Newtonian Mechanics . . . . . . . . . . . . . . . . . . 1.1.1 The equation of motion for a single particle . . 1.1.2 Angular Motion . . . . . . . . . . . . . . . . . 1.1.3 Energy and Work . . . . . . . . . . . . . . . . . 1.2 Gravitation . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Gravitational Force . . . . . . . . . . . . . . . . 1.2.2 Gravitational Potential . . . . . . . . . . . . . 1.3 Dynamics of Systems of Particles . . . . . . . . . . . . 1.3.1 Newtonian Mechanical Concepts for Systems of 1.3.2 The Virial Theorem . . . . . . . . . . . . . . . 1.3.3 Collisions of Particles . . . . . . . . . . . . . . 1 2 2 13 16 24 24 26 32 32 47 51 63 64 65 71 76 81 84 85 86 86 92 96 103 103 108 109 119 123 123 130 138 138 147 ...... ...... ...... ...... ...... ...... ...... ...... Particles ...... ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Lagrangian and Hamiltonian Dynamics 2.1 The Lagrangian Approach to Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Degrees of Freedom, Constraints, and Generalized Coordinates . . . . . . . 2.1.2 Virtual Displacement, Virtual Work, and Generalized Forces . . . . . . . . 2.1.3 d’Alembert’s Principle and the Generalized Equation of Motion . . . . . . . 2.1.4 The Lagrangian and the Euler-Lagrange Equations . . . . . . . . . . . . . . 2.1.5 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Cyclic Coordinates and Canonical Momenta . . . . . . . . . . . . . . . . . . 2.1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 More examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.9 Special Nonconservative Cases . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.10 Symmetry Transformations, Conserved Quantities, Cyclic Coordinates and Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Variational Calculus and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Variational Calculus and the Euler Equation . . . . . . . . . . . . . . . 2.2.2 The Principle of Least Action and the Euler-Lagrange Equation . . . . . . 2.2.3 Imposing Constraints in Variational Dynamics . . . . . . . . . . . . . . . . 2.2.4 Incorporating Nonholonomic Constraints in Variational Dynamics . . . . . 2.3 Hamiltonian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Legendre Transformations and Hamilton’s Equations of Motion . . . . . . . 2.3.2 Phase Space and Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . 2.4 Topics in Theoretical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Canonical Transformations and Generating Functions . . . . . . . . . . . . 2.4.2 Symplectic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . CONTENTS 2.4.3 2.4.4 2.4.5 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Action-Angle Variables and Adiabatic Invariance . . . . . . . . . . . . . . . . 152 The Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 160 173 . 174 . 174 . 176 . 177 . 181 . 186 . 191 . 191 . 195 . 204 . 211 . 215 . 215 . 219 . 222 . 229 233 . 234 . 234 . 240 . 243 . 243 . 248 . 252 . 252 . 253 . 255 257 258 258 260 262 263 267 269 269 278 280 282 282 296 3 Oscillations 3.1 The Simple Harmonic Oscillator . . . . . . . . . . . . . . . . 3.1.1 Equilibria and Oscillations . . . . . . . . . . . . . . . 3.1.2 Solving the Simple Harmonic Oscillator . . . . . . . . 3.1.3 The Damped Simple Harmonic Oscillator . . . . . . . 3.1.4 The Driven Simple and Damped Harmonic Oscillator 3.1.5 Behavior when Driven Near Resonance . . . . . . . . . 3.2 Coupled Simple Harmonic Oscillators . . . . . . . . . . . . . 3.2.1 The Coupled Pendulum Example . . . . . . . . . . . . 3.2.2 General Method of Solution . . . . . . . . . . . . . . . 3.2.3 Examples and Applications . . . . . . . . . . . . . . . 3.2.4 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Loaded String . . . . . . . . . . . . . . . . . . . . 3.3.2 The Continuous String . . . . . . . . . . . . . . . . . . 3.3.3 The Wave Equation . . . . . . . . . . . . . . . . . . . 3.3.4 Phase Velocity, Group Velocity, and Wave Packets . . 4 Central Force Motion and Scattering 4.1 The Generic Central Force Problem . . . . . . . . . . . . 4.1.1 The Equation of Motion . . . . . . . . . . . . . . . 4.1.2 Formal Implications of the Equations of Motion . . 4.2 The Special Case of Gravity – The Kepler Problem . . . . 4.2.1 The Shape of Solutions of the Kepler Problem . . 4.2.2 Time Dependence of the Kepler Problem Solutions 4.3 Scattering Cross Sections . . . . . . . . . . . . . . . . . . 4.3.1 Setting up the Problem . . . . . . . . . . . . . . . 4.3.2 The Generic Cross Section . . . . . . . . . . . . . . 4.3.3 1 Potentials . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Rotating Systems 5.1 The Mathematical Description of Rotations . . . . . . . . . . . . . . . 5.1.1 Inﬁnitesimal Rotations . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Finite Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Interpretation of Rotations . . . . . . . . . . . . . . . . . . . . 5.1.4 Scalars, Vectors, and Tensors . . . . . . . . . . . . . . . . . . . 5.1.5 Comments on Lie Algebras and Lie Groups . . . . . . . . . . . 5.2 Dynamics in Rotating Coordinate Systems . . . . . . . . . . . . . . . . 5.2.1 Newton’s Second Law in Rotating Coordinate Systems . . . . . 5.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Lagrangian and Hamiltonian Dynamics in Rotating Coordinate 5.3 Rotational Dynamics of Rigid Bodies . . . . . . . . . . . . . . . . . . . 5.3.1 Basic Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Torque-Free Motion . . . . . . . . . . . . . . . . . . . . . . . . iv ..... ..... ..... ..... ..... ..... ..... ..... ..... Systems ..... ..... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS 5.3.3 Motion under the Inﬂuence of External Torques . . . . . . . . . . . . . . . . . 313 323 . 324 . 324 . 324 . 333 . 339 . 346 347 347 348 349 354 356 357 357 359 359 363 371 379 384 389 6 Special Relativity 6.1 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 The Postulates . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Transformation Laws . . . . . . . . . . . . . . . . . . . 6.1.3 Mathematical Description of Lorentz Transformations 6.1.4 Physical Implications . . . . . . . . . . . . . . . . . . . 6.1.5 Lagrangian and Hamiltonian Dynamics in Relativity . A Mathematical Appendix A.1 Notational Conventions for Mathematical Symbols . . A.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . A.3 Vector and Tensor Deﬁnitions and Algebraic Identities A.4 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . A.5 Taylor Expansion . . . . . . . . . . . . . . . . . . . . . A.6 Calculus of Variations . . . . . . . . . . . . . . . . . . A.7 Legendre Transformations . . . . . . . . . . . . . . . . B Summary of Physical Results B.1 Elementary Mechanics . . . . . . . . . . . . B.2 Lagrangian and Hamiltonian Dynamics . . B.3 Oscillations . . . . . . . . . . . . . . . . . . B.4 Central Forces and Dynamics of Scattering B.5 Rotating Systems . . . . . . . . . . . . . . . B.6 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Chapter 1 Elementary Mechanics This chapter reviews material that was covered in your ﬁrst-year mechanics course – Newtonian mechanics, elementary gravitation, and dynamics of systems of particles. None of this material should be surprising or new. Special emphasis is placed on those aspects that we will return to later in the course. If you feel less than fully comfortable with this material, please take the time to review it now, before we hit the interesting new stuﬀ! The material in this section is largely from Thornton Chapters 2, 5, and 9. Small parts of it are covered in Hand and Finch Chapter 4, but they use the language of Lagrangian mechanics that you have not yet learned. Other references are provided in the notes. 1 CHAPTER 1. ELEMENTARY MECHANICS 1.1 Newtonian Mechanics References: • Thornton and Marion, Classical Dynamics of Particles and Systems, Sections 2.4, 2.5, and 2.6 • Goldstein, Classical Mechanics, Sections 1.1 and 1.2 • Symon, Mechanics, Sections 1.7, 2.1-2.6, 3.1-3.9, and 3.11-3.12 • any ﬁrst-year physics text Unlike some texts, we’re going to be very pragmatic and ignore niceties regarding the equivalence principle, the logical structure of Newton’s laws, etc. I will take it as given that we all have an intuitive understanding of velocity, mass, force, inertial reference frames, etc. Later in the course we will reexamine some of these concepts. But, for now, let’s get on with it! 1.1.1 The equation of motion for a single particle We study the implications of the relation between force and rate of change of momentum provided by Newton’s second law. Deﬁnitions Position of a particle as a function of time: r(t) Velocity of a particle as a function of time: v(t) = the velocity, v = |v|, as the speed. d dt r(t). We refer to the magnitude of d dt Acceleration of a particle as a function of time: a(t) = Momentum of a particle: p(t) = m(t) v(t) Newton’s second law In inertial frames, it holds that F (t) = If the mass is not time-dependent, we have F (t) = m d p(t) dt v(t) = d2 dt2 r(t). (1.1) d d2 v(t) = m 2 r(t) dt dt d dt (1.2) ˙ We use the “dot” shorthand, deﬁning r = ¨ r and r = d2 dt2 r, which gives (1.3) ˙ ˙ ¨ F = p = m v = mr Newton’s second law provides the equation of motion, which is simply the equation that needs to be solved ﬁnd the position of the particle as a function of time. Conservation of Linear Momentum: Suppose the force on a particle is F and that there is a vector s such that the force has no component along s; that is F ·s=0 (1.4) 2 1.1. NEWTONIAN MECHANICS ˙ Newton’s second law is F = p, so we therefore have ˙ p · s = 0 =⇒ p · s = α (1.5) where α is a constant. That is, there is conservation of the component of linear momentum along the direction s in which there is no force. Solving simple Newtonian mechanics problems Try to systematically perform the following steps when solving problems: • Sketch the problem, drawing all the forces as vectors. • Deﬁne a coordinate system in which the motion will be convenient; in particular, try to make any constraints work out simply. • Find the net force along each coordinate axis by breaking down the forces into their components and write down Newton’s second law component by component. • Apply the constraints, which will produce relationships among the diﬀerent equations (or will show that the motion along certain coordinates is trivial). • Solve the equations to ﬁnd the acceleration along each coordinate in terms of the known forces. • Depending on what result is desired, one either can use the acceleration equations directly or one can integrate them to ﬁnd the velocity and position as a function of time, modulo initial conditions. • If so desired, apply initial conditions to obtain the full solution. Example 1.1 (Thornton Example 2.1) A block slides without friction down a ﬁxed, inclined plane. The angle of the incline is θ = 30◦ from horizontal. What is the acceleration of the block? • Sketch: Fg = mg is the gravitational force on the block and FN is the normal force, which is exerted by the plane on the block to keep it in place on top of the plane. • Coordinate system: x pointing down along the surface of the incline, y perpendicular to the surface of the incline. The constraint of the block sliding on the plane forces there to be no motion along y, hence the choice of coordinate system. 3 CHAPTER 1. ELEMENTARY MECHANICS • Forces along each axis: m x = Fg sin θ ¨ m y = FN − Fg cos θ ¨ • Apply constraints: there is no motion along the y axis, so y = 0, which gives FN = ¨ Fg cos θ. The constraint actually turns out to be unnecessary for solving for the motion of the block, but in more complicated cases the constraint will be important. • Solve the remaining equations: Here, we simply have the x equation, which gives: x= ¨ Fg sin θ = g sin θ m where Fg = mg is the gravitational force • Find velocity and position as a function of time: This is just trivial integration: t d x = g sin θ =⇒ x(t) = x(t = 0) + ˙ ˙ ˙ dt g sin θ dt 0 = x0 + g t sin θ ˙ d x = x(t = 0) + g t sin θ =⇒ x(t) = x0 + ˙ dt t dt x0 + g t sin θ ˙ 0 = x0 + x0 t + ˙ 12 g t sin θ 2 where we have taken x0 and x0 to be the initial position and velocity, the constants of ˙ integration. Of course, the solution for y is y(t) = 0, where we have made use of the initial conditions y(t = 0) = 0 and y(t = 0) = 0. ˙ Example 1.2 (Thornton Example 2.3) Same as Example 1.1, but now assume the block is moving (i.e., its initial velocity is nonzero) and that it is subject to sliding friction. Determine the acceleration of the block for the angle θ = 30◦ assuming the frictional force obeys Ff = µk FN where µk = 0.3 is the coeﬃcient of kinetic friction. • Sketch: We now have an additional frictional force Ff which points along the −x direction because the block of course wants to slide to +x. Its value is ﬁxed to be Ff = µk FN . 4 1.1. NEWTONIAN MECHANICS • Coordinate system: same as before. • Forces along each axis: m x = Fg sin θ − Ff ¨ m y = FN − Fg cos θ ¨ We have the additional frictional force acting along −x. • Apply constraints: there is no motion along the y axis, so y = 0, which gives FN = ¨ Fg cos θ. Since Ff = µk FN , the equation resulting from the constraint can be used directly to simplify the other equation. • Solve the remaining equations: Here, we simply have the x equation, x= ¨ Fg Fg sin θ − µk cos θ m m = g [sin θ − µk cos θ] That is all that was asked for. For θ = 30◦ , the numerical result is x = g (sin 30◦ − 0.3 cos 30◦ ) = 0.24 g ¨ Example 1.3 (Thornton Example 2.2) Same as Example 1.1, but now allow for static friction to hold the block in place, with coeﬃcient of static friction µs = 0.4. At what angle does it become possible for the block to slide? • Sketch: Same as before, except the distinction is that the frictional force Ff does not have a ﬁxed value, but we know its maximum value is µs FN . • Coordinate system: same as before. • Forces along each axis: m x = Fg sin θ − Ff ¨ m y = FN − Fg cos θ ¨ • Apply constraints: there is no motion along the y axis, so y = 0, which gives FN = ¨ Fg cos θ. We will use the result of the application of the constraint below. • Solve the remaining equations: Here, we simply have the x equation, x= ¨ Ff Fg sin θ − m m • Since we are solving a static problem, we don’t need to go to the eﬀort of integrating to ﬁnd x(t); in fact, since the coeﬃcient of sliding friction is usually lower than the coeﬃcient of static friction, the above equations become incorrect as the block begins to move. Instead, we want to ﬁgure out at what angle θ = θ the block begins to slide. Since Ff has maximum value µs FN = µs m g cos θ, it holds that x≥ ¨ 5 Fg FN sin θ − µs m m CHAPTER 1. ELEMENTARY MECHANICS i.e., x ≥ g [sin θ − µs cos θ] ¨ It becomes impossible for the block to stay motionless when the right side becomes positive. The transition angle θ is of course when the right side vanishes, when 0 = sin θ − µs cos θ or tan θ which gives θ = 21.8◦ . Atwood’s machine problems Another class of problems Newtonian mechanics problems you have no doubt seen before are Atwood’s machine problems, where an Atwood’s machine is simply a smooth, massless pulley (with zero diameter) with two masses suspended from a (weightless) rope at each end and acted on by gravity. These problems again require only Newton’s second equation. Example 1.4 (Thornton Example 2.9) Determine the acceleration of the two masses of a simple Atwood’s machine, with one ﬁxed pulley and two masses m1 and m2 . • Sketch: = µs • Coordinate system: There is only vertical motion, so use the z coordinates of the two masses z1 and z2 . 6 1.1. NEWTONIAN MECHANICS • Forces along each axis: Just the z-axis, but now for two particles: m1 z1 = −m1 g + T ¨ m2 z2 = −m2 g + T ¨ where T is the tension in the rope. We have assumed the rope perfectly transmits force from one end to the other. • Constraints: The ro..."

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