1 Introduction
1.1 Background of elastica theory
1.2 The goal of this book
1.3 A brief review of the classical elastica in three-dimensionalEuclidean space E3
1.4 A simple introduction to elliptic functions and elliptic equations 13
1.5 Some basic concepts of affine space
2 The Centroaffine elastica in the affine plane R2
2.1 Definition and variation formula of the generalized centroaffineelastica in R2
2.2 Integration of the generalized centroaffine elastica in R2
2.3 Centro-affine elastica for p(k) = k2 + X in the affine plane R2
2.3.1 The solutions of motion equation for centroaffine elastica
2.3.2 The solutions of the structure equation for centroaffine elastica in the affine plane R2
2.3.3 Closed centroaffine elastica
2.3.4 Second variation formula
3 Centroaffine elastica in affine space R3
3.1 Definition and variation formula of the energy functional in R3 59
3.2 Integration of the generalized centroaffine elastica in R3
3.3 Centroaffine elastica for p(k) = k2 + A in the affine space R3
3.3.1 Motion equation of the centroaffine elastica
3.3.2 Solutions of the centroaffine elastica
3.3.3 Closed centroaffine elastica
3.4 The variation of centroaffine torsion for Starlike curves
3.5 Hamiltonian lows on the space of starlike curves in the affine space R3
3.6 Flow Generated by the vector field -k'X+kX'
4 Affine elastica in the affine plane R2
4.1 Definition and variation formula of affine elastica in the affine plane R2
4.2 Affine elastica for p(k) = k2 + X in the affine plane R2
4.2.1 The solutions of motion equation for affine elastica
4.2.2 Second variation of affine elastica in R2
4.3 Critical curves for p(k) = k + > in the affine plane R2
5 Affine elastica in the affine space R3
5.1 Definition and variation formula of the affine elastica in theaffine space R3
5.2 Variation of the energy functional Jx(k2 + A)ds in the affinespace R3
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