  There is a remarkable connection between the clique number and the Lagrangian
of a 2-graph proved by Motzkin and Straus in 1965. It is useful in practice if
similar results hold for hypergraphs. However the obvious generalization of
Motzkin and Straus' result to hypergraphs is false. Frankl and F\"{u}redi
conjectured that the $r$-uniform hypergraph with $m$ edges formed by taking the
first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest
Lagrangian of all $r$-uniform hypergraphs with $m$ edges. For $r=2$, Motzkin
and Straus' theorem confirms this conjecture. For $r=3$, it is shown by Talbot
that this conjecture is true when $m$ is in certain ranges. In this paper, we
explore the connection between the clique number and Lagrangians for
$3$-uniform hypergraphs. As an application of this connection, we confirm that
Frankl and F\"{u}redi's conjecture holds for bigger ranges of $m$ when $r$=3.
We also obtain two weaker versions of Tur\'{a}n type theorem for
left-compressed $3$-uniform hypergraphs.
