A well-known conjecture of Frankl and F"{u}redi from 1989 states that an initial segment of the colexicographic order has the largest Lagrangian of any $r$-uniform hypergraph with $m$ hyperedges. We show that this is true when $r=3$. We also give a new proof of a related conjecture of Nikiforov for large $t$ and a counterexample to an old conjecture of Ahlswede and Katona.