Worm-like chain under external forces: Some analytic results

Exact results of the classical mechanical solution of the worm-like chain model under an external stretching force are derived. The locus of the chain under an arbitrary stretching force and grafting conditions is calculated explicitly in terms of elliptic functions. In the long chain limit, the chain locus can be expressed in terms of elementary functions. In addition, the solution of a worm-like rod chain under a small twisting torque is also calculated perturbatively. The relative extension, writhe number, and the locus of the chain are computed to leading orders in the reduced torque.