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Lemons D.S. — Perfect form: Variational principles, methods, and applications in elementary physics
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 physics

Автор: 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.
The text gradually unfolds the physics and mathematics. While other treatments postulate Hamilton's principle and deduce all results from it, Perfect Form begins with the most plausible and restricted variational principles and develops more powerful ones through generalization. One selection of text and problems even constitutes a non-calculus of variations introduction to variational methods, while the mathematics more generally employed extends only to solving simple ordinary differential equations. Perfect Form is designed to supplement existing classical mechanics texts and to present variational principles and methods to students who approach the subject for the first time.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 128

Добавлена в каталог: 01.12.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
"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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