  The dynamics for a thin, closed loop inextensible Euler's elastica moving in
three dimensions are considered. The equations of motion for the elastica
include a wave equation involving fourth order spatial derivatives and a second
order elliptic equation for its tension. A Hasimoto transformation is used to
rewrite the equations in convenient coordinates for the space and time
derivatives of the tangent vector. A feature of this elastica is that it
exhibits time-dependent monodromy. A frame parallel-transported along the
elastica is in general only quasi-periodic, resulting in time-dependent
boundary conditions for the coordinates. This complication is addressed by a
gauge transformation, after which a contraction mapping argument can be
applied. Local existence and uniqueness of elastica solutions are established
for initial data in suitable Sobolev spaces.
