Ebook: Exterior Differential Systems and the Calculus of Variations

15 0. PRELIMINARIES a) Notations from Manifold Theory b) The Language of Jet Manifolds c) Frame Manifolds d) Differentia! Ideals e) Exterior Differential Systems EULER-LAGRANGE EQUATIONS FOR DIFFERENTIAL SYSTEMS ~liTH ONE I. 32 INDEPENDENT VARIABLE a) Setting up the Problem; Classical Examples b) Variational Equations for Integral Manifolds of Differential Systems c) Differential Systems in Good Form; the Derived Flag, Cauchy Characteristics, and Prolongation of Exterior Differential Systems d) Derivation of the Euler-Lagrange Equations; Examples e) The Euler-Lagrange Differential System; Non-Degenerate Variational Problems; Examples FIRST INTEGRALS OF THE EULER-LAGRANGE SYSTEM; NOETHER'S II. 1D7 THEOREM AND EXAMPLES a) First Integrals and Noether's Theorem; Some Classical Examples; Variational Problems Algebraically Integrable by Quadratures b) Investigation of the Euler-Lagrange System for Some Differential-Geometric Variational Pro~lems: 2 i) ( K ds for Plane Curves; i i) Affine Arclength; 2 iii) f K ds for Space Curves; and iv) Delauney Problem. II I. EULER EQUATIONS FOR VARIATIONAL PROBLEfiJS IN HOMOGENEOUS SPACES 161 a) Derivation of the Equations: i) Motivation; i i) Review of the Classical Case; iii) the Genera 1 Euler Equations 2 K /2 ds b) Examples: i) the Euler Equations Associated to f for lEn; but for Curves in i i) Some Problems as in i) sn; Non- Curves in iii) Euler Equations Associated to degenerate Ruled Surfaces IV.

---

---

Content:
Front Matter....Pages i-ix
Introduction....Pages 1-14
Preliminaries....Pages 15-31
Euler-Lagrange Equations for Differential Systems with One Independent Variable....Pages 32-106
First Integrals of the Euler-Lagrange System; Noether’s Theorem and Examples....Pages 107-160
Euler Equations for Variational Problems in Homogeneous Spaces....Pages 161-198
Endpoint Conditions; Jacobi Equations and the 2nd Variation; the Hamilton-Jacobi Equation....Pages 199-309
Back Matter....Pages 310-339

---

Content:
Front Matter....Pages i-ix
Introduction....Pages 1-14
Preliminaries....Pages 15-31
Euler-Lagrange Equations for Differential Systems with One Independent Variable....Pages 32-106
First Integrals of the Euler-Lagrange System; Noether’s Theorem and Examples....Pages 107-160
Euler Equations for Variational Problems in Homogeneous Spaces....Pages 161-198
Endpoint Conditions; Jacobi Equations and the 2nd Variation; the Hamilton-Jacobi Equation....Pages 199-309
Back Matter....Pages 310-339
....