  We investigated the apsidal motion for the multi-planet systems. In the
simulations, we found that the two planets of HD 37124, HD 12661, 47 Uma and HD
82943 separately undergo apsidal alignment or antialignment. But the companions
of GJ 876 and $\upsilon$ And are only in apsidal lock about $0^{\circ}$.
Moreover, we obtained the criteria with Laplace-Lagrange secular theory to
discern whether a pair of planets for a certain system are in libration or
circulation.
