.

0 (hodnocen0 x )

(3) Půjčeno:6x 

BK

Mineola : Dover Publications, 2000

viii, 232 s. ; 23 cm

ISBN 978-0-486-41448-5 (brož.)

Dover books on mathematics

ruština

Obsahuje bibliografii na s. 227 a rejstřík

000053113

CONTENTS // ELEMENTS OF THE THEORY Page 1. 1: Functionals. Some Simple Variational Problems, 1. 2: Function Spaces, 4. 3: The Variation of a Functional. A // Necessary Condition for an Extremum, 8. 4: The Simplest Variational Problem. Euler’s Equation, 14. 5: The // Case of Several Variables, 22. 6: A Simple Variable // End Point Problem, 25. 7: The Variational Derivative, // 27. 8: Invariance of Euler’s Equation, 29. Problems, // 31. // FURTHER GENERALIZATIONS Page 34. 9: The // Fixed End Point Problem for n Unknown Functions, 34. // 10. Variational Problems in Parametric Form, 38. 11: // Functionals Depending on Higher-Order Derivatives, 40. // 12: Variational Problems with Subsidiary Conditions, 42. // Problems, 50. // THE GENERAL VARIATION OF A FUNCTIONAL // Page 54. 13: Derivation of the Basic Formula, 54. 14: // End Points Lying on Two Given Curves or Surfaces, 59. // 15: Broken Extremals. The Weierstrass-Erdmann Conditions, 61. Problems, 63. // THE CANONICAL FORM OF THE EULER EQUATIONS AND RELATED TOPICS Page 67. 16: The // Canonical Form of the Euler Equations, 67. 17: First // Integrals of the Euler Equations, 70. 18: The Legendre // Transformation, 71. 19: Canonical Transformations, 77. // 20: Noether’s Theorem, 79. 21: The Principle of Least // Action, 83. 22: Conservation Laws, 85. 23: The Hamilton-Jacobi Equation. Jacobi’s Theorem, 88. Problems, // 94. // vi CONTENTS // THE SECOND VARIATION. SUFFICIENT CONDITIONS FOR A WEAK EXTREMUM Page

97. 24: // Quadratic Functionals. The Second Variation of a Functional, 97. 25: The Formula for the Second Variation. // Legendre’s Condition, 101. 26: Analysis of the Quadcb // ratic Functional (Ph’2 4- Qh2) dx 105. 27: Jacobi’s // J a // Necessary Condition. More on Conjugate Points, 111. // 28: Sufficient Conditions for a Weak Extremum, 115. // 29: Generalization to n Unknown Functions, 117. 30: // Connection Between Jacobi’s Condition and the Theory of // Quadratic Forms, 125. Problems, 129. // FIELDS. SUFFICIENT CONDITIONS FOR A STRONG // EXTREMUM Page 131. 31: Consistent Boundary Conditions. General Definition of a Field, 131. 32: The // Field of a Functional, 137. 33: Hilbert’s Invariant Integral, 145. 34: The Weierstrass E-Function. Sufficient // Conditions for a Strong Extremum, 146. Problems, 150. // VARIATIONAL PROBLEMS INVOLVING MULTIPLE // INTEGRALS Page 152. 35: Variation of a Functional // Defined on a Fixed Region, 152. 36: Variational Derivation of the Equations of Motion of Continuous Mechanical Systems, 154. 37: Variation of a Functional // Defined on a Variable Region, 168. 38: Applications to // Field Theory, 180. Problems, 190. // DIRECT METHODS IN THE CALCULUS OF VARIATIONS Page 192. 39: Minimizing Sequences, 193. // 40: The Ritz Method and the Method of Finite Differences, 195. 41: The Sturm-Liouville Problem, 198. // Problems, 206. // APPENDIX I // PROPAGATION OF DISTURBANCES AND THE CANONICAL EQUATIONS Page 208. // CONTENTS

VÜ // APPENDIX II VARIATIONAL METHODS IN PROBLEMS OF OPTIMAL CONTROL Page 218. // BIBLIOGRAPHY Page 227. // INDEX Page 228.