  We study variational systems for space curves, for which the Lagrangian or
action principle has a Euclidean symmetry, using the Rotation Minimising frame,
also known as the Normal, Parallel or Bishop frame.
  Such systems have previously been studied using the Frenet-Serret frame. The
Rotation Minimising frame has many advantages, however, and can be used to
study a wider class of examples.
  We achieve our results by extending the powerful symbolic invariant calculus
for Lie group based moving frames, to the Rotation Minimising frame case. To
date, the invariant calculus has been developed for frames defined by algebraic
equations. By contrast, the Rotation Minimising frame is defined by a
differential equation.
  In this paper, we derive the recurrence formulae for the symbolic invariant
differentiation of the symbolic invariants. We then derive the syzygy operator
needed to obtain Noether's conservation laws as well as the Euler-Lagrange
equations directly in terms of the invariants, for variational problems with a
Euclidean symmetry. We show how to use the six Noether laws to ease the
integration problem for the minimising curve, once the Euler-Lagrange equations
have been solved for the generating differential invariants. Our applications
include variational problems used in the study of strands of proteins, nucleid
acids and polymers.
