Volume 155, number 2,3 PHYSICS LETTERS A 6 May 1991 Topological lower bound on the energy of a twisted rod John C. Baez Department ofMathematics and Computer Science, University of California, Riverside, CA 92521, USA and Rossen Dandoloff Department of Physics, University of California, Riverside, CA 92521, USA Received 8 June 1990; revised manuscript received 19 February 1991; accepted for publication 28 February 1991 Communicated by A.R. Bishop If one end of an elastic rod is rotated by an angle of 2,~ relative to the other, the “body frame” along the rod traces Out a noncontractible loop in SO(3). This is not the case for a rotation by 47r. A lower bound is derived for the energy of a thin elastic rod whose body frame traces out a noncontractible loop in S0(3). If one takes an elastic rod, holds one end fixed, and L twists the other through an angle of 2it, the twist can- ~ ,~ ~ j,~ ds (1) not be undone by moving either end as long as the orientations of the ends are fixed. However, if one twists by an angle of 4it, the twist can be undone by (under the approximations made in ref. [31),where moving the ends of the rods holding their orienta- w is the tangent vector dF/ds, and I, are the prin- tions fixed. This is because the rotation group in three cipal moments of inertia: I~ and ‘2 for the cross-sec- dimensions, SO (3), is doubly connected. Here we tion of the rod and 13 for the torsional rigidity of the use this fact to derive lower bounds on the energy of rod. In particular, 11= ‘2 for a homogeneous rod with a thin elastic rod with one end twisted by an angle a circular cross-section. of 2ir. While there have been a number of applica- As a digression, note that if one interprets the pa- tions of topology to continuum mechanics [1,21, this rameter s in eq. (1) as time, then E equals the action rather simple result seems not to have been noted for the time evolution of a rigid body with moments before, of inertia I~ and angular velocity w. Thus the prob- The state of a thin elastic rod may be described by lem of the thin elastic rod may be mapped onto the a function F from the interval [0, L], where L is the time evolution of a rotating rigid body. This was ap- length of the rod, to SO(3). For each point se [0, LI, parently first noted by Kirchhoff [4]. F(s) describes the “body frame” of the rod as ro- Give SO (3) the Riemannian metric g such that tated from the standard frame (e 1, e2, e3). We may identify a tangent vector w at any point xa SO(3) with a vector (a~, a2, w3) in the Lie algebra 1k011 2 ~ ~ 1=1 so(3) P3 by left translation of the tangent space at x to the identity in SO(3). The elastic energy of the cf. ref. [5]. Let g 0 denote this metric in the special rod is then given by case where I~ = 1 for all i. Note that g>~ Imjngo, where ‘mm denotes the minimum of the I,. Using this and the Cauchy—Schwarz inequality we have 0375-9601/91/1 03.50 © 1991 — Elsevjer Science Publishers B.V. (North-Holland) 145