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Lemons D.S. — Perfect form: Variational principles, methods, and applications in elementary physics
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Название: Perfect form: Variational principles, methods, and applications in elementary physicsАвтор: Lemons D.S. Аннотация: What does the path taken by a ray of light share with the trajectory of a thrown baseball and the curve of a wheat stalk bending in the breeze? Each is the subject of a different study yet all are optimal shapes; light rays minimize travel time while a thrown baseball minimizes action. All natural curves and shapes, and many artificial ones, manifest such "perfect form" because physical principles can be expressed as a statement requiring some important physical quantity to be mathematically maximum, minimum, or stationary. Perfect Form introduces the basic "variational" principles of classical physics (least time, least potential energy, least action, and Hamilton's principle), develops the mathematical language most suited to their application (the calculus of variations), and presents applications from the physics usually encountered in introductory course sequences.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1997Количество страниц: 128Добавлена в каталог: 01.12.2013Операции: Положить на полку |
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"Falling light" 41 "Least" principles versus "stationary" principles 9 "Natural" place 48 action 70 Action, multiparticle 102 Aristotelean causes x 12—13 Aristotle 45 48 Bernoulli, Jacob 23 Bernoulli, Johann 23 Brachistochrone 28—29 Calculus of variations 17 23 Calculus of variations, first problem of 20 Calculus of variations, Fundamental Lemma of 23 27 Cantilever model 62—63 Catenary 55—56 Catenary, symmetric 64 Causes, Aristotelian 12—13 Center of mass coordinates 103 Comparison set 19 Compound pendulum 113 Conservative system 96 Constants of motion 89 Constraints on mechanical systems 50 Constraints on mechanical systems, holonomic 106 Constraints on mechanical systems, isopermetric 56 Constraints on mechanical systems, nonholonomic 106 Coordinates, Cartesian 26 Coordinates, center of mass 103 Coordinates, cylindrical polar 26 Coordinates, generalized 26 106—107 Coordinates, ignorable 28 Coordinates, other coordinates 90 Coordinates, spherical polar 26 Cycloid 29 de Fermat, Pierre 3—5 7 13 de Maupertuis, Pierre Louis Moreau 13 69 89 de Maupertuis, Pierre Louis Moreau, discovery of least action 71 Descartes, Rene 7 8 Dido 56 Dido, problem of 64—66 Direct variational method 19 Direct variational method versus Euler — Lagrange 30—31 Duffing's equation 42 Effective potential 105 Efficient cause 12 Elastic column 58—61 Elastic pendulum 92—93 Elasticity, modulus of 59 Electrostatic energy 63 Ellipsoid of revolution 114 Energy conservation in the restricted Hamilton's Principle 84—86 Energy conservation, lack of in the extended Hamilton's Principle 96 Equilibria 47 Equilibria, oscillations around 92—94 Euler — Lagrange equation 22—23 27—28 Euler — Lagrange equation, first integrals of 24—25 Euler, Leonhard x 13 17 23 72 89 Fermat's principle ix x 3 5 7 8—9 10 12 17 19 33 69 see Fermat's Principle, equivalence to Least Action 69 76—77 fiber optic 37 Fiber optic, helical rays 38 43 Fiber optic, meridional rays 38 42 Fiber optic, skew rays 38 Final cause x 12—13 First integrals of the Euler — Lagrange equation 24—25 First integrals of the Euler — Lagrange equation, relation to symmetry 24 Focal length 11 Foucault, Leon 7 Foucault, Leon, measurement of the speed of light 7 71 Functional 20 82 Fundamental lemma of the calculus of variations 23 27 Galileo 47 Generalized coordinates 90—91 106—108 Generalized Snell's Law 34 36 41 Geodesic 73 77 Geometrical optics 3—5 Geometrical optics, focal length 11 Geometrical optics, image formation 9—12 Geometrical optics, paraxial ray approximation 10 42 Geometrical optics, spherical abberations 11 Geometrical optics, thin lens approximation 9—10 Gladestone — Dale Law 35 Hamilton's first principal function 82 96 Hamilton's Principle versus Newton's Laws 87 102 Hamilton's principle, extended 96 Hamilton's Principle, restricted 81—83 Hamilton, William Rowan 72 83 90 95 Hamilton, William Rowan, on Lagrange 89 Hamiltonian systems 96 Hamiltonian systems, multiparticle 100—102 Harmonic motion 92 harmonic oscillator 113 Helical rays 38 43 Hero of Alexandria 13 Hero's Problem ix 13 Holonomic constraints 106 Hooke's law 49 Huygens, Christian 33 Hydrostatic balance 50—52 54 Ignorable coordinate 28 Image formation 10—12 Index of refraction, absolute 6 Index of refraction, relative 3 Inertial reference frame 47 Isopermetric constraints 56 Jacobi's Principle of Least Action 72—73 Jacobi's Principle of Least Action, relation to Fermat's Principle 76—77 Jacobi, C.G.J. 72 73 Kepler's third law 92 Kepler's Third Law, generalized 89 92—93 L'Hospital 23 Lagrange multipliers 52—54 Lagrange's equations of motion 86 Lagrange, Joseph Louis 23 72 81 86 89 Lagrange, Joseph Louis, contribution to variational dynamics 89 Lagrange, Joseph Louis, Hamilton on 89 Lagrange, Joseph Louis, Mechanique Analtyic 89 Lagrangian 86 Lagrangian in natural form 96 Lagrangian, multiparticle 101 112—113 Least resistance 7 15 Leibniz, Gottfried Wilhelm ix 7 12 13 23 Light rays and particle trajectories 77 Light rays, helical 38 43 Light rays, meridional 38 Light rays, relation to particle trajectories 77 Light rays, skew 38 Light speed, measured by Foucault 7 Light speed, measured by Roemer 4 Line elements, Cartesian 25 Line elements, cylindrical polar 26 Line elements, spherical polar 26 Loaded beam 66 Luneberg lens 43 Mach, Ernst 13 Meridional rays 38 42 Metric 25 Mirages 34—37 Multiparticle action 102 Multiparticle Hamiltonian systems 100—102 Multiparticle Lagrangian 102 112 Natural boundary conditions 56—58 Newton's Second Law of Motion 47 Newton's Second Law of Motion from Hamilton's Principle 87 Newton's Second Law of Motion versus Hamilton's Principle 87 102 Newton, Isaac 7 23 Newton, Isaac, his Principia 47 Optical path length 33 Orbit shapes 78 Oscillations around equilibria 92—94 Oscillations around equilibria of elastic pendulum 89 92—93 Parametric ray equations 39 Paraxial ray approximation 10 42 Particle trajectories and light rays 77 Pascal, Blaise 52 Pascal, Blaise, his hydrostatic principle 52 Pendulum, elastic 89 92—93 Pendulum, spherical 78 87—89 Principia 47 Principles, Variational, Fermat's see "Least Time" Principles, Variational, of Least Action 69—72 73 Principles, Variational, of Least Potential Energy 45—47 87 Principles, Variational, of Least Time 3 5 7 8—9 12 17 33 69 76 Principles, Variational, of Stationary Action 71 Principles, Variational, of Stationary Potential Energy 47 Principles, Variational, of Stationary Time 9 Projectile trajectory 73—75 Ray optics 3—5 7 see Reflection, law of 3 Roemer, Olaf 4 Skew rays 38 Snell's law 3 5—6 Snell's Law, generalized 34 36 41 Speed of light, measured by Foucault 7 Speed of light, measured by Roemer 4 Spherical abberations 11 Spherical mirror 15 Spherical pendulum 78 87—89 Stability of mechanical systems 48 Stability of mechanical systems of driven Watt's governor 99—100 Stability of mechanical systems of loaded flywheel 61—62 Stability of mechanical systems of undriven Watt's governor 111—112 Symmetry, relation to first integrals 24 Thin lens approximation 9—10 True rays 7 19 Two-body problem 103—106 Variational principles xii see Variational principles versus least principles 9 Watt's governor, driven 97—100 Watt's governor, undriven 111—112 Watt, James 97 Young's constant 59
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