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Lauwerier H.A. — Calculus of variations in mathematical physics
Lauwerier H.A. — Calculus of variations in mathematical physics

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Название: Calculus of variations in mathematical physics

Автор: Lauwerier H.A.


This tract represents worked-out lecture notes of a course in the calculus of variations delivered by the author to students in mathematical physics at the University of Amsterdam. In this course the calcucalculus of variations is treated in a slightly modernized way by making full use of the language of vector spaces. Although the reader is supposed to be familiar with the fundamental notions of a Banach space and a Hilbert space, two sections are included in which these spaces are treated systematically in a condensed fashion. Much attention is paid to problems of theoretical mechanics including Noether's theorem. Some elementary knowledge of boundary value problems, e.g. vibrating string and membrane, will enable the reader to appreciate more fully those parts of the text, in which applications of Hilbert space theory are made. Much material for this course is derived from the books by Gelfand and Fomin and by Michlin. In particular, the first book represents an easily readable modern introduction to the calculus of variations and its applications.

Язык: en

Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1966

Количество страниц: 106

Добавлена в каталог: 19.07.2005

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Предметный указатель
Arzela — Ascoli criterion      38
Banach space, definition of      35
Banach space, functions on      41—43
Bernoulli, James      20
Bernoulli, John      20
Bessel, inequality of      46
Brachistochrone problem      20—24 61 68
Catenary problem      25—28
Direct methods      89—98
Eigenvalue problem      91 92
Euler's equations      16—19
Euler's equations, brachistochrone problem      22 23
Euler's equations, catenary problem      26
Euler's equations, functions of higher order      58
Euler's equations, geodesic      55
Euler, L.      20
Extremals, broken      24 81—83
Extremals, definition of      16
Fourier series, coefficients of      45
Fourier series, general case of      90
Fourier series, special case of      33 34
Functional(s), bilinear      38
Functional(s), linear      38
Functional(s), of higher order      58 59
Functional(s), on a Banach space      41—43
Functional(s), on a Hilbert space      49—53
Functional(s), quadratic      38
Functional(s), twice differentiable      42
Geodesies      55 56
Hamilton's principle      56 57 67—71
Hilbert space      44—48
Hilbert space, definition of      45
Inner product      44
Kepler's second law      57 88
L'Hospital      20
Lagrange equation of motion      68 70
Lagrange multipliers      7 8
Lagrange multipliers for catenary problem      26 27
Laplace equation      65 66
Least action, principle of      69
Legendre, condition of      32
Legendre, theorem of      32
Linear normed space      35—40
Linear normed space, definition of      35
Lower semi-continuous sfunctional      94
Minimal surface      22—24
Natural boundary conditions      60 61 65
Neighbouring curve      15
Newton, I.      20
Noether's theorem      84—88
Norm, definition of      35
Operator, linear      49—53
Operator, symmetric      50
Orthogonal complements      44
Orthogonal elements      44
Orthogonal set      44
Parseval relation      45 46
Pendulum      71
Quadratic form      5 6
Riess, F., theorem of      47 48
Saddle points      5
Second variation      29—34
Sequence, convergence of      35—37
Sequence, fundamental      35
Sequence, minimizing      93—98
Set (in linear normed space), compact, definition of      37
Set (in linear normed space), equicontinuous      37 38
Stationary points, definition of      2
Torricelli, point of      9
Variable end points      60 61
Variation, first      15—17
Variation, first, definition of      30 41
Variation, general      85
Variation, general, definition of      38
Variation, second      29—34
Variation, second, definition of      30 43
Variational problem (see contents), for more than one variable      62—66
Variational problem (see contents), simplest      15—19
Vibrating membrane      78 79 90
Vibrating rod      79 80
Vibrating string      75—78 92
Weierstrass theorem      1 39
Weierstrass — Erdmann condition      82 83
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