By Phillip A. Griffiths (auth.)
15 zero. PRELIMINARIES a) Notations from Manifold concept b) The Language of Jet Manifolds c) body Manifolds d) Differentia! beliefs e) external Differential structures EULER-LAGRANGE EQUATIONS FOR DIFFERENTIAL structures ~liTH ONE I. 32 self sustaining VARIABLE a) developing the matter; Classical Examples b) Variational Equations for quintessential Manifolds of Differential platforms c) Differential platforms in stable shape; the Derived Flag, Cauchy features, and Prolongation of external Differential platforms d) Derivation of the Euler-Lagrange Equations; Examples e) The Euler-Lagrange Differential method; Non-Degenerate Variational difficulties; Examples FIRST INTEGRALS OF THE EULER-LAGRANGE method; NOETHER'S II. 1D7 THEOREM AND EXAMPLES a) First Integrals and Noether's Theorem; a few Classical Examples; Variational difficulties Algebraically Integrable by way of Quadratures b) research of the Euler-Lagrange method for a few Differential-Geometric Variational Pro~lems: 2 i) ( okay ds for aircraft Curves; i i) Affine Arclength; 2 iii) f ok ds for area Curves; and iv) Delauney challenge. II I. EULER EQUATIONS FOR VARIATIONAL PROBLEfiJS IN HOMOGENEOUS areas 161 a) Derivation of the Equations: i) Motivation; i i) assessment of the Classical Case; iii) the Genera 1 Euler Equations 2 okay /2 ds b) Examples: i) the Euler Equations linked to f for lEn; yet for Curves in i i) a few difficulties as in i) sn; Non- Curves in iii) Euler Equations linked to degenerate governed Surfaces IV.