  Quasisymmetric stellarators are appealing intellectually and as fusion
reactor candidates since the guiding center particle trajectories and
neoclassical transport are isomorphic to those in a tokamak, implying good
confinement. Previously, quasisymmetric magnetic fields have been identified by
applying black-box optimization algorithms to minimize symmetry-breaking
Fourier modes of the field strength $B$. Here instead we directly construct
magnetic fields in cylindrical coordinates that are quasisymmetric to leading
order in distance from the magnetic axis, without using optimization. The
method involves solution of a 1-dimensional nonlinear ordinary differential
equation, originally derived by Garren and Boozer [Phys. Fluids B 3, 2805
(1991)]. We demonstrate the usefulness and accuracy of this optimization-free
approach by providing the results of this construction as input to the codes
VMEC and BOOZ_XFORM, confirming the purity and scaling of the magnetic
spectrum. The space of magnetic fields that are quasisymmetric to this order is
parameterized by the magnetic axis shape along with three other real numbers,
one of which reflects the on-axis toroidal current density, and another one of
which is zero for stellarator symmetry. The method here could be used to
generate good initial conditions for conventional optimization, and its speed
enables exhaustive searches of parameter space.
