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15 zero. PRELIMINARIES a) Notations from Manifold concept b) The Language of Jet Manifolds c) body Manifolds d) Differentia! beliefs e) external Differential platforms EULER-LAGRANGE EQUATIONS FOR DIFFERENTIAL platforms ~liTH ONE I. 32 self sustaining VARIABLE a) establishing the matter; Classical Examples b) Variational Equations for indispensable Manifolds of Differential structures c) Differential structures in reliable shape; the Derived Flag, Cauchy features, and Prolongation of external Differential structures d) Derivation of the Euler-Lagrange Equations; Examples e) The Euler-Lagrange Differential procedure; Non-Degenerate Variational difficulties; Examples FIRST INTEGRALS OF THE EULER-LAGRANGE procedure; 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 process 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 house 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) overview of the Classical Case; iii) the Genera 1 Euler Equations 2 ok /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.
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