  Starting from the vortex filament flow introduced in 1906 by Da Rios, there
is a hierarchy of commuting geometric flows on space curves. The traditional
approach relates those flows to the nonlinear Schr\"odinger hierarchy satisfied
by the complex curvature function of the space curve. Rather than working with
this infinitesimal invariant, we describe the flows directly as vector fields
on the manifold of space curves. This manifold carries a canonical symplectic
form introduced by Marsden and Weinstein. Our flows are precisely the
symplectic gradients of a natural hierarchy of invariants, beginning with
length, total torsion, and elastic energy. There are a number of advantages to
our geometric approach. For instance, the real part of the spectral curve is
geometrically realized as the motion of the monodromy axis when varying total
torsion. This insight provides a new explicit formula for the hierarchy of
Hamiltonians. We also interpret the complex spectral curve in terms of curves
in hyperbolic space and Darboux transforms. Furthermore, we complete the
hierarchy of Hamiltonians by adding area and volume. These allow for the
characterization of elastic curves as solutions to an isoperimetric problem:
elastica are the critical points of length while fixing area and volume.
