Elastica in SO(3)

Abstract

In a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group 50(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = 50(3), we solve the Euler-Lagrange equation by quadratures.2000 Mathematics subject classification: primary 49K99,49Q99.

Global Solutions of the Equation of the Kirchhoff Elastic Rod in Space Forms

“…Since γ is unit-speed, the three vectors T , ∇ T T and (∇ T ) 2 T along γ must satisfy some conditions. (The argument is similar to the case of elasticae; see [37].) First, |T | = 1 holds.…”

Extendability of Kirchhoff elastic rods in complete Riemannian manifolds

“…(The argument is similar to the case of elasticae; see Theorem 1.2 of Ref. 48.) First, |γ | = 1 holds.…”

Magnetic trajectories in three‐dimensional Lie groups

“…If a matrix differential equation can be written in this form, then the spectrum of V is preserved by the flow. 11 In the present situation, the Lax equation .1), we obtain the following equalities:…”

“…Differential equations of the form are called Lax equations. If a matrix differential equation can be written in this form, then the spectrum of V is preserved by the flow . In the present situation, the Lax equation is crucial to the solution of or equivalently for a magnetic trajectory γ in terms of its Lie reduction V .…”

“…respectively. 10,11 Now, we suppose that is a reparametrized curve by arc length s in G. Let t (s) = d (s) ds = · (s) denote the unit tangent vector field to , n(s) the unit normal vector field and b(s) the binormal vector field. Then, {t, n, b} is the Frenet frame for with formulas…”