  We study the spin-orbit coupling in spin-triplet Cooper pairs and clarify
multiple superconducting (SC) phases in Sr$_2$RuO$_4$. Based on the analysis of
the three-orbital Hubbard model with atomic LS coupling, we show some selection
rules of the spin-orbit coupling in Cooper pairs. The spin-orbit coupling is
small when the two-dimensional $\gamma$-band is the main cause of the
superconductivity, although the LS coupling is much larger than the SC gap.
Considering this case, we investigate multiple SC transitions in the magnetic
fields for both H // [001] and H // [100] using the Ginzburg-Landau theory and
the quasi-classical theory. Rich phase diagrams are obtained because the spin
degree of freedom in Cooper pairs is not quenched by the spin-orbit coupling.
Experimental indications for the multiple phases in Sr$_2$RuO$_4$ are
discussed.
