  We study stationary points of the bending energy of curves
$\gamma\colon[a,b]\to\mathbb{R}^n$ subject to constraints on the arc-length and
total torsion while simultaneously allowing for a variable bending stiffness
along the arc-length of the curve. Physically, this can be understood as a
model for an elastic wire with isotropic cross-section of varying thickness. We
derive the corresponding Euler-Lagrange equations for variations that are
compactly supported away from the end points thus obtaining characterizations
for elastic curves with variable bending stiffness. Moreover, we provide a
collection of alternative characterizations, e.g., in terms of the curvature
function. Adding to numerous known results relating elastic curves to dynamics,
we establish connections between elastic curves with variable bending stiffness
and damped pendulums and the flow of vortex filaments with finite thickness.
