Classical Mechanics. for the 19th century. T. Helliwell V. Sahakian

Transcription

1 Classical Mechanics for the 19th century T. Helliwell V. Sahakian

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3 Contents 1 Newtonian particle mechanics Inertial frames and the Galilean transformation Newton s laws of motion Example 1-1: A bacterium with a viscous drag force Example 1-2: A linearly damped oscillator 1.3 Systems of particles Conservation laws Example 1-3: A wrench in space Example 1-4: A particle moving in two dimensions with an attractive spring force Example 1-5: Particle in a magnetic field Example 1-6: A child on a swing Example 1-7: A particle attached to a spring revisited Example 1-8: Newtonian central gravity and its potential energy Example 1-9: Dropping a particle in spherical gravity Example 1-10: Potential energies and turning points for positive powerlaw forces 1.5 Forces of Nature Dimensional analysis Example 1-11: Find the rate at which molasses flows through a narrow pipe 1.7 Synopsis Relativity Foundations The Postulates The Lorentz transformation Example 2-1: Rotation and rapidity i

4 ii CONTENTS 2.2 Relativistic kinematics Proper time Four-velocity Example 2-2: The transformation of ordinary velocity Example 2-3: Four-velocity invariant 2.3 Relativistic dynamics Four-momentum Example 2-4: Relativistic dispersion relation Example 2-5: Decay into two particles Four-force Dynamics in practice Example 2-6: Uniformly accelerated motion Example 2-7: The Doppler effect Minkowski diagrams Example 2-8: Time dilation Example 2-9: Length contraction Example 2-10: The twin paradox 3 The Variational Principle Fermat s principle The calculus of variations Geodesics Example 3-1: Geodesics on a plane Example 3-2: Geodesics on a sphere 3.4 Brachistochrone Example 3-3: Fermat again 3.5 Several Dependent Variables Example 3-4: Geodesics in three dimensions 3.6 Mechanics from a variational principle Motion in a uniform gravitational field Summary Lagrangian mechanics The Lagrangian in Cartesian coordinates Hamilton s principle Example 4-1: A simple pendulum Example 4-2: A bead sliding on a vertical helix Example 4-3: Block on an inclined plane

5 CONTENTS iii 4.3 Generalized momenta and cyclic coordinates Example 4-4: Particle on a tabletop, with a central force Example 4-5: The spherical pendulum 4.4 Systems of particles Example 4-6: Two interacting particles Example 4-7: Pulleys everywhere Example 4-8: A block on a movable inclined plane 4.5 The Hamiltonian Example 4-9: Bead on a rotating parabolic wire 4.6 The moral of constraints Small oscillations about equilibrium Example 4-10: Particle on a tabletop with a central spring force 4.8 Relativistic generalization Summary Appendices 187 Appendix A When is H E? Beyond The Basics I 5.1 Classical waves Example 5-1: Two-slit interference of waves 5.2 Two-slit experiments with light and atoms Feynman sum-over-paths Two slits and two paths No barriers at all Example 5-2: A class of paths near a straight-line path Example 5-3: How classical is the path? 5.6 Path shapes for light rays and particles Example 5-4: Path shape for particles in uniform gravity 5.7 Why Hamilton s principle? Conclusions Symmetries and Conservation Laws Cyclic coordinates and generalized momenta Example 6-1: A star orbiting a spheroidal galaxy Example 6-2: A charged particle moving outside a charged rod 6.2 A less straightforward example

6 iv CONTENTS 6.3 Infinitesimal transformations Example 6-3: Translations Example 6-4: Rotations Example 6-5: Lorentz transformations 6.4 Symmetry Noether s theorem Example 6-6: Space translations and momentum Example 6-7: Time translation and the Hamiltonian Example 6-8: Rotations and angular momentum Example 6-9: Galilean Boosts Example 6-10: Lorentz invariance Example 6-11: Sculpting Lagrangians from symmetry 6.6 Some comments on symmetries Gravitation and Central-force motion Central forces The two-body problem The effective potential energy Radial motion for the central-spring problem Radial motion in central gravity The shape of central-force orbits Central spring-force orbits The shape of gravitational orbits Example 7-1: Orbital geometry and orbital physics 7.5 Bertrand s Theorem Orbital dynamics Kepler s second law Kepler s third law Example 7-2: Halley s Comet Minimum-energy transfer orbits Example 7-3: A voyage to Mars Example 7-4: Gravitational assists 8 Electromagnetism The Lorentz force law Example 8-1: Fixing a gauge 8.2 The Lagrangian for electromagnetism The two-body problem, once again

7 CONTENTS v 8.4 Coulomb scattering Example 8-2: Snell scattering 8.5 Uniform magnetic field Example 8-3: Bubble chamber Example 8-4: Ion trapping 8.6 Contact forces Example 8-5: A microscopic model Example 8-6: Rolling down the plane Example 8-7: Stacking barrels Example 8-8: On the rope 9 Accelerating frames Linearly accelerating frames Example 9-1: Pendulum in an accelerating spaceship 9.2 Rotating frames Example 9-2: Throwing a ball in a rotating space colony Example 9-3: Polar orbits around the Earth 9.3 Pseudoforces in rotating frames Centrifugal and Coriolis pseudoforces Example 9-4: Rotating space colonies revisited 9.5 Pseudoforces on Earth Example 9-5: Coriolis pseudoforces in airflow Example 9-6: Foucault s pendulum 9.6 Spacecraft rendezvous and docking Example 9-7: Rendezvous with the space station? Example 9-8: Losing a wrench? 10 Beyond The Basics II 10.1 Beyond newtonian gravity Example 10-1: The precession of Mercury s perihelion Magnetic gravity Example 10-2: Gravity inside the body of a star Example 10-3: Cosmic string 10.2 Beyond the classical forces Beyond deterministic forces

8 vi CONTENTS 11 Hamiltonian formulation Legendre transformations Example 11-1: A simple Legendre transform 11.2 Hamilton s equations Phase Space Example 11-2: The simple harmonic oscillator Example 11-3: A bead on a parabolic wire Example 11-4: A charged particle in a uniform magnetic field 11.4 Canonical transformations Example 11-5: Transforming the simple harmonic oscillator Example 11-6: Identities Example 11-7: Infinitesimal transformations and the Hamiltonian Example 11-8: Point transformations 11.5 Poisson brackets Example 11-9: Position and momenta Example 11-10: The simple harmonic oscillator once again 11.6 Liouville s theorem Rigid Body Dynamics Rigid Bodies Rotations Infinitesimal Rotations Example 12-1: Rotations in higher dimensions 12.4 The Euler Angles Example 12-2: Angular velocity transformation 12.5 Rotational kinetic energy Example 12-3: A hoop 12.6 Potential Energy Angular Momentum Torque Summary Principal Axes Example 12-4: Fixed Axis Rotation Example 12-5: Principal Axis Shifts 12.11Torque Free Dynamics Example 12-6: Adding Angular Momenta 12.12Gyroscopes

9 CONTENTS vii 13 Complex systems Chaos Small oscillations Beyond The Basics III 15.1 Beyond classical phase space

10 viii CONTENTS

11 List of Figures 1.1 Various inertial frames in space. If one of these frames is inertial, any other frame moving at constant velocity relative to it is also inertial Two inertial frames, O and O, moving relative to one another along their mutual x or x axes A bacterium in a fluid. What is its motion if it begins with velocity v 0 and then stops swimming? Motion of an oscillator if it is (a) overdamped, (b) underdamped, or (c) critically damped, for the special case where the oscillator is released from rest (v 0 = 0) at some position x A system of particles, with each particle identified by a position vector r A collection of particles, each with a position vector r i from a fixed origin. The center of mass R CM is shown, and also the position vector r i of the ith particle measured from the center of mass The position vector for a particle. Angular momentum is always defined with respect to a chosen point from where the position vector originates A two-dimensional elliptical orbit of a ball subject to a Hooke s law spring force, with one end of the spring fixed at the origin The work done by a force on a particle is its line integral along the path traced by the particle Newtonian gravity pulling a probe mass m towards a source mass M ix

12 x LIST OF FIGURES 1.11 Potential energy functions for selected positive powers n. A possible energy E is drawn as a horizontal line, since E is constant. The difference between E and U(x) at any point is the value of the kinetic energy T. The kinetic energy is zero at the turning points, where the E line intersects U(x). Note that for n = 1 there are two turning points for E > 0, but for n = 2 there is only a single turning point Inertial frames O and O Graph of the γ factor as a function of the relative velocity β. Note that γ = 1 for nonrelativistic particles, and γ as β The velocity v x as a function of v x for fixed relative frame velocity V = 0.5c A particle of mass m 0 decays into two particles with masses m 1 and m 2. Both energy and momentum are conserved in the decay, but mass is not conserved in relativistic physics. That is, m 0 m 1 + m Plots of relativistic constant-acceleration motion. (a) shows v x (t), demonstrating that v x (t) c as t, i.e., the speed of light is a speed limit in Nature. The dashed line shows the incorrect Newtonian prediction. (b) shows the hyperbolic trajectory of the particle on a c t-x graph. Once again the dashed trajectory is the Newtonian prediction Observer O shooting a laser towards observer O while moving towards O A point on a Minkowski diagram represents an event. A particle s trajectory appears as a curve with a slope that exceeds unity everywhere Three events on a Minkowski diagram. Events A and B are timelike separated; A and C are lightlike separated; and B and C are spacelike separated The hyperbolic trajectory of a particle undergoing constant acceleration motion on a Minkowski diagram The grid lines of two observers labeling the same event on a spacetime Minkowski diagram

13 LIST OF FIGURES xi 2.11 The time dilation phenomenon. (a) Shows the scenario of a clock carried by observer O. (b) shows the case of a clock carried by O The phenomenon of length contraction. (a) Shows the scenario of a meter stick carried by observer O. (b) shows the case of a stick carried by O Minkowski diagrams of the twin paradox. (a) shows simultaneity lines according to John. During the first and third part of the trip, a time 2 T 1 elapse on John s clock; during the middle part, Jane is accelerating uniformly and the time elapsed is denoted by T 0 (b) Shows simultaneity lines according to Jane, except for the two dotted lines sandwiching the accelerating segment. Jane s x axis is also shown for two instants in time. The segment labeled T 0 is excised away and borrowed from John s perspective since Jane is a not an inertial frame during this period. T 1 and T 2 on the other hand can be computed from Jane s perspective. Notice how Jane s x axis must smoothly flip around during the time interval T 0, as she turns around. Her simultaneity lines during this period will hence be distorted and require general relativity to fully unravel Light traveling by the least-time path between a and b, in which it moves partly through air and partly through a piece of glass. At the interface the relationship between the angle θ 1 in air, with index of refraction n 1, and the angle θ 2 in glass, with index of refraction n 2, is n 1 sin θ 1 = n 2 sin θ 2, known as Snell s law. This phenomenon is readily verified by experiment A light ray from a star travels down through Earth s atmosphere on its way to the ground A function of two variables f(x 1, x 2 ) with a local minimum at point A, a local maximum at point B, and a saddle point at C Various paths y(x) that can be used as input to the functional I[f(x)]. We look for that special path from which an arbitrary small displacement δy(x) leaves the functional unchanged to linear order in δy(x). Note that δy(a) = δy(b) = A discretization of a path The coordinates θ and ϕ on a sphere

14 xii LIST OF FIGURES 3.7 (a) Great circles on a sphere are geodesics; (b) Two paths nearby the longer of the two great-circle routes of a path Possible least-time paths for a sliding block A cycloid. If in darkness you watch a wheel rolling along a level surface, with a lighted bulb attached to a point on the outer rim of the wheel, the bulb will trace out the shape of a cycloid. In the diagram the wheel is rolling along horizontally beneath the surface. For x b < (π/2)y b, the rail may look like the segment from a to b 1 ; for x b > (π/2)y b, the segment from a to b 2 would be needed (a) A light ray passing through a stack of atmospheric layers; (b) The same problem visualized as a sequence of adjacent slabs of air of different index of refraction Two spaceships, one accelerating in gravity-free space (a), and the other at rest on the ground (b). Neither observers in the accelerating ship nor those in the ship at rest on the ground can find out which ship they are in on the basis of any experiments carried out solely within their ship A laser beam travels from the bow to the stern of the accelerating ship Cartesian, cylindrical, and spherical coordinates A bead sliding on a vertically-oriented helical wire Block sliding down an inclined plane Particle moving on a tabletop The effective radial potential energy for a mass m moving with an effective potential energy U eff = (p ϕ ) 2 /2mr 2 + (1/2)kr 2 for various values of p ϕ, m, and k Coordinates of a ball hanging on an unstretchable string A sketch of the effective potential energy U eff for a spherical pendulum. A ball at the minimum of U eff is circling the vertical axis passing through the point of suspension, at constant θ. The fact that there is a potential energy minimum at some angle θ 0 means that if disturbed from this value the ball will oscillate back and forth about θ 0 as it orbits the vertical axis Two interacting beads on a one-dimensional frictionless rail. The interaction between the particles depends only on the distance between them

15 LIST OF FIGURES xiii 4.9 A contraption of pulleys. We want to find the accelerations of all three weights. We assume that the pulleys have negligible mass so they have negligible kinetic and potential energies A block slides along an inclined plane. Both block and inclined plane are free to move along frictionless surfaces A bead slides without friction on a vertically-oriented parabolic wire that is forced to spin about its axis of symmetry The effective potential U eff for the Hamiltonian of a bead on a rotating parabolic wire with z = αr 2, depending upon whether the angular velocity ω is less than, greater than, or equal to ω crit = 2g α An effective potential energy U eff with a focus near a minimum. Such a point is a stable equilibrium point. The dotted parabola shows the leading approximation to the potential near its minimum. As the energy drains out, the system settles into its minimum with the final moments being well approximated with harmonic oscillatory dynamics The shape of the two-dimensional orbit of a particle subject to a central spring force, for small oscillations about the equilibrium radius A transverse small displacement of a string (a) A small slice of string; (b) Tension forces on the slice Two paths for waves from slit system to detectors (a) The relationship between s 2 s 1, d, and θ; (b) The two-slit interference pattern (a) At very low intensity light, individual photons appear to land on the screen randomly; (b) as the intensity is cranked up, the interference pattern emerges

16 xiv LIST OF FIGURES 5.6 Helium atoms with speeds between 2.1 and 2.2 km/s reaching the rear detectors, with both slits open. The detectors observe the arrival of individual atoms, but the distribution shows a clear interference pattern as we would expect for waves!. We see how the interference pattern builds up one atom at a time. The first data set is taken after 5 minutes of counting, while the last is taken after 42 hours of counting. The experiments were carried out by Ch. Kurtsiefer, T. Pfau, and J. Mlynek; see their article in Nature 386, 150 (1997). (The hotspot in the data arises from an enhanced dark count due to an impurity in the microchannel plate detector.) A phasor z(t 0 )e iφ z(t 0 ) e i(φ+φ 0) drawn in the complex plane. The real axis is horizontal and the imaginary axis is vertical. The absolute length of the phasor is z(t 0 ) and the angle between the phasor and the real axis is the phase (φ+φ 0 ), where φ 0 is the phase of z(t 0 ) alone The sum of two individual phasors with the same magnitudes z(t 0 ) but different phases. The result is a phasor that extends from the tail of the first to the tip of the second, as in vector addition. The difference in their angles in the complex plane is the difference in their phase angles. Shown are examples with phase differences equal to (a) zero (b) 45 (c) 90 (d) 135 (e) Two paths from a source to a detector High-velocity helium atoms, with speeds above 30 km/s, reaching the rear detectors, with both slits open. The detectors observe the arrival of individual atoms, and the distribution is what we would expect for classical particles. Experiments carried out by Ch. Kurtsiever, T. Pfau, and J. Mlynek, Nature 386, (a) Path length as a function of position y within the slit. (b) The single-slit diffraction pattern The double slit, with a screen at distance D. We can view the intensity on the screen as a function of the transverse distance x

17 LIST OF FIGURES xv 5.13 Interference/diffraction patterns for a double slit with a = d/4 and D = 1000d. The diffraction curves, shown in dashed lines, serve as envelopes for the more rapidly oscillating interference pattern. (a) The pattern in the case d = 0.1x 1/2, where x 1/2 is the distance on the detecting plane between the center and the first minimum of the diffraction envelope. The diffraction curves of the two slits strongly overlap in this case, giving in effect a single diffraction envelope. (b) The pattern in the case d = 2x 1/2, showing that the two diffraction patterns have become separated, with the first minimum due to each slit at the same location in the center. This case corresponds to a wavelength smaller by a factor of 20 than the pattern shown in (a) The sum of a large number of phasors (a) that are about the same (b) that differ by constant amounts A class of kinked paths between a source and detector. The straight line is the shortest path, and the midpoint of the others is a distance D = n D 0 from the straight line, where (n = ±1, ±2,...) Phasors up to n = ±25. The more distant paths wind up in spirals, contributing very little to the overall phasor sum An elliptical galaxy (NGC 1132) pulling on a star at the outer fringes The two types of transformations considered: direct on the left, indirect on the right Sliding pendulum Newtonian gravity pulling a probe mass m 2 towards a source mass m Angular momentum conservation and the planar nature of central force orbits The classical two-body problem in physics The effective potential for the central-spring potential The effective gravitational potential Elliptical orbits due to a central spring force F = kr Conic sections: circles, ellipses, parabolas, and hyperbolas

18 xvi LIST OF FIGURES 7.8 An elliptical gravitational orbit, showing the foci, the semimajor axis a, semiminor axis b, the eccentricity ɛ, and the periapse and apoapse Parabolic and hyperbolic orbits The four types of gravitational orbits The area of a thin pie slice The orbit of Halley s comet A minimum-energy transfer orbit to an outer planet Insertion from a parking orbit into the transfer orbit A spacecraft flies by Jupiter, in the reference frames of (a) Jupiter (b) the Sun The electrostatic Coulomb force between two charged particles Hyperbolic trajectory of a probe scattering off a charged target Definition of the scattering cross section in terms of change in impact area 2πbdb and scattering area 2π sin ΘdΘ on the unit sphere centered at the target The Rutherford scattering cross section. The graph shows log σ(θ) as a function of log Θ superimposed on actual data in scattering of protons off gold atoms Scattering of light off a reflecting bead (a) Top view of a charged particle in a uniform magnetic field; (b) The helical trajectory of the charged particle The effective Penning potential. At the minimum, we have a stable circular trajectory. In general however, the radial extent will oscillate with frequency ω The full trajectory of an ion in a Penning trap. A vertical oscillation along the z axis with frequency ω z is superimposed onto an fast oscillation of frequency ω 0, while the particle traces a large circle with characteristic frequency ω m The electric field from a neutral atom leaks out in a dipole pattern due to small asymmetries in the charge distribution of the atom A layer of perfectly aligned dipole at the surface of a floor on which a block is to rest A hoop rolling down an inclined plane without slipping

19 LIST OF FIGURES xvii 8.13 Two barrels stacked on top of each other. The lower barrel is stationary, while the upper one rolls down without slipping A pendulum with a single constraint given by the fixed length of the rope A ball is thrown sideways in an accelerating spaceship (a) as seen by observers within the ship (b) as seen by a hypothetical inertial observer outside the ship A simple pendulum in an accelerating spaceship spacecolony living on the inside rim of a rotating cylindrical space colony Throwing a ball in a rotating space colony (a) From the point of view of an external inertial observer (b) From the point of view of a colonist Path of a satellite orbiting Earth, in Earth s rest frame. Dashed lines represent the equator and longitude lines A vector that is constant in a rotating frame changes in an inertial frame: (a) simple two dimensional case; (b) three dimensional general case (a) The angular velocity vector for a rotating frame (b) the triple cross product ω ω r Stroboscopic pictures of a ball thrown from the center of a rotating space colony (a) as seen in an inertial frame (b) as seen in the colony The length of the day relatively to the stars (sidereal time) is slightly longer than the length of the day relative to the Sun The Earth bulges at the equator due to its rotation, which produces a centrifugal pseudoforce in the rotating frame. A plumb bob hanging near the surface experiences both gravitation and the centrifugal pseudoforce (a) A set of three Cartesian coordinates placed on the Earth (b) The horizontal coordinates x and y Inflowing air develops a counterclockwise rotation in the northern hemisphere Foucault s pendulum at the North Pole Foucault s pendulum

20 xviii LIST OF FIGURES 9.15 (a) A spacecraft trying to rendezvous and dock with a space station in circular orbit around the Earth. (b) A stranded astronaut trying to return to the space station by throwing a wrench. (c) An astronaut accidentally lets a wrench escape from the ISS. What is its subsequent trajectory? Coordinates of the space station and object The spacecraft trajectory in the nonrotating frame Rendezvous with the ISS? The bizarre trajectory, after starting off in the desired direction Rendezvous with the ISS? The initial boost Trajectory of a wrench in the rotating frame in which the ISS is at rest. The wrench is thrown from the ISS vertically, away from the Earth. It returns like a boomerang Trajectory of the wrench in the nonrotating frame where the ISS is in circular orbit around the Earth (a) a balloon in a car (b) a cork in a fishtank Tilt of the northward-flowing gulf stream surface, looking north An ant colony measures the radius and circumference of a turntable Non-Euclidean geometry: circumferences on a sphere Successive light rays sent to a clock at altitude h from a clock on the ground Effective potential for the Schwarzschild geometry A gaussian surface probing the gravity inside a star of uniform volume mass density An infinite linear mass distribution moves upward with speed V while a probe of mass M ventures nearby (a) Two functions A(x, y), differing by a shift, whose naive transformation through y z lead to the same transformed function B(x, z); (b) The envelope of A(x, y) consisting of slopes and intercepts completely describe the shape of A(x, y) The Legendre transformation of A(x, y) as B(x, z) The two dimensional cross section of a phase space for a system. The flow lines depict Hamiltonian time evolution The phase space of the one dimensional simple harmonic oscillator

21 LIST OF FIGURES xix 11.5 The phase space of the one dimensional particle on a parabola problem The flow lines in the x-p x cross section of phase space for a charged particle in a uniform magnetic field (a) The flow lines in a given phase space; (b) The same flow lines as described by transformation coordinates and momenta The transformation of phase space under a canonical transformation. Volume elements may get distorted in shape, but the volume of each element must remain unchanged A depiction of Liouville s theorem: the density of states of a system evolves in phase space in such as way that its total time derivative is zero

22 xx LIST OF FIGURES

23 LIST OF FIGURES 1