  We prove local in time well-posedness for a large class of quasilinear
Hamiltonian, or parity preserving, Schr\"odinger equations on the circle. After
a paralinearization of the equation, we perform several paradifferential
changes of coordinates in order to transform the system into a paradifferential
one with symbols which, at the positive order, are constant and purely
imaginary. This allows to obtain a priori energy estimates on the Sobolev norms
of the solutions.
