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.
