The calculus of variations

Francis D Murnaghan

Table of contents :
Title page……Page 1
Date-line……Page 2
Series……Page 3
Preface……Page 4
Contents……Page 5
1. The Lagrangian Function and the Parametric Integrand……Page 6
2. Extremal Curves; The Euler-Lagrange Equation……Page 12
3. Lagrangian Functions Which are Linear in $x_t$……Page 18
4. The Legendre Condition for a Minimal Curve……Page 22
5. Proof of the Legendre Condition……Page 26
6. Constrained Problems: The Hamilton Canonical Equations……Page 31
7. The Reciprocity between $L$ and $H$; The Transversality Conditions……Page 36
8. Extremal Fields; The Hilbert Invariant Integral……Page 41
9. The Weierstrass $E$-Function; Positively Regular Problems……Page 46
10. A Simple Example of the Construction of an Extremal Field; Rayleigh Quotients and the Method of Rayleigh-Ritz……Page 51
11. The Principle of Maupertuis; The Propagation of Waves……Page 57
12. Problems Whose Lagrangian Functions Involve Derivatives of Higher Order than the First……Page 65
13. Multiple-Integral Problems of the Calculus of Variations……Page 75
14. Constrained Problems; Characteristic Numbers……Page 80
15. Multiple-Integral Problems Whose Lagrangian Functions Involve Derivatives of Higher Order than the First……Page 85
16. The Courant Maximum-Minimum Principle……Page 92
Bibliography……Page 99
Index……Page 100