The calculus of variations

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