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.
General
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