  We present the zero temperature phase diagram of a Heisenberg antiferromagnet
on a frustrated triangular lattice with nearest neighbor ($J_1$) and next
nearest neighbor ($J_2$) interactions, in a magnetic field. We show that the
classical model has an accidental degeneracy for all $J_2/J_1$ and all fields,
but the degeneracy is lifted by quantum fluctuations. We show that at large
$S$, for $J_2/J_1 <1/8$, quantum fluctuations select the same sequence of three
sublattice co-planar states in a field as for $J_2 =0$, and for $1/8<J_2/J_1
<1$ they select the canted stripe state for all non-zero fields. The transition
between the two states is first order in all fields, with the hysteresis width
set by quantum fluctuations. We study the model with arbitrary $S$, including
$S=1/2$, near the saturation field by exploring the fact that near saturation
the density of bosons is small for all $S$. We show that for $S >1$, the
transition remains first order, with a finite hysteresis width, but for $S=1/2$
and, possibly, $S=1$, there appears a new intermediate phase, likely without a
spontaneous long-range order.
