GEOMETRICALLY CONSTRAINED CURVES AND THE CALCULUS OF VARIATIONS

[摘要] A physical wire in a simply connected 3-dimensional complete Riemannian manifold M with constant sectional curvature resists bending and twisting. If we clamp the ends of a wire in some position then it will assume the shape of a certain curve and twist along that curve so as to minimize its total energy. Let (gamma): a,b (--->) M be a curve representing the position vector of a wire. Let (gamma)(t) = ((gamma)(t), e(,1)(t), e(,2)(t), e(,3)(t)) where e(,1)(t) is the unit tangent vector of (gamma)(t); (e(,1)(t), e(,2)(t), e(,3)(t)) is an oriented orthonormal frame of T(,(gamma)(t))M. Then we can define a real-valued smooth function (sigma)(t) > 0, m(,1)(t), m(,2)(t), m(,3)(t) as follows: (UNFORMATTED TABLE FOLLOWS) d(gamma)/dt = (sigma)e(,1) de(,1)/dt = -m(,3)(sigma)e(,2) + m(,2)(sigma)e(,3) de(,2)/dt = m(,3)(sigma)e(,1)-m(,1)(sigma)e(,3) de(,3)/dt = -m(,2)(sigma)e(,1) + m(,1)(sigma)e(,2)(TABLE ENDS) Let constants A, B represent how much the wire resisting bending and twisting. Let AI(,1)((gamma)) = A (INT) 1/2(m(,2)(;;2) + m(,3)(;;2))ds represent the total bending energy stored in the wire while BI(,2) ((gamma)) = B (INT) 1/2 m(,1)(;;2)ds represents the total twisting energy stored in the wire. We assume that the total energy stored in the wire is I((gamma)) = AI(,1) ((gamma)) + BI(,2) ((gamma)). In this thesis we investigate the global behavior of the curve (gamma) so as to minimize its total energy I((gamma)). We use exterior differential system techniques to study Euler-Lagrange systems. We find that the system is completely integrable and in the generic case the lifting problem reduces to integrating a linear flow on a torus.

[发布机构] Rice University