  We present a unified approach for determining and proving obstructions to
small-time local controllability of scalar-input control systems. Our approach
views obstructions to controllability as resulting from interpolation
inequalities between the functionals associated with the formal Lie brackets of
the system.
  Using this approach, we give compact unified proofs of all known necessary
conditions, we prove a conjecture of 1986 due to Kawski, and we derive entirely
new obstructions. Our work doubles the number of previously-known necessary
conditions, all established in the 1980s. In particular, for the third
quadratic bracket, we derive a new necessary condition which is complementary
to the Agrachev-Gamkrelidze sufficient ones.
  We rely on a recent Magnus-type representation formula for the state, a new
Hall basis of the free Lie algebra over two generators, an appropriate use of
Sussmann's infinite product to compute the Magnus expansion, and
Gagliardo-Nirenberg interpolation inequalities.
