Slender, elastic rod-like structures on or inside constrained rigid surfaces are prevalent in a wide range of engineering (drill strings in borewells, pipelines under the seabed, ocean cables), medical (stents in angioplasty of arteries), biological (DNA toroidal condensates, bacterial flagella), electronic (carbon nanotubes) and robotic (soft robots for in-pipe inspection) applications. This project seeks to employ a comprehensive variational theory of elastic two-strand braids to investigate the post-buckling behaviour of elastic rods lying on rigid tubular surfaces. Methods comprising the calculus of variations and Lagrangian and Hamiltonian mechanics are utilised to procure more general types of solutions to various nonlinear boundary value problems, using both analytical and numerical approaches.