לזכות ר' יהודה יעקב בן דינה חאשע
ור' משה יהודה בן פעשא דינה
In the last mishna of tomid we read the list of all of the shir shel yom's that we say daily after davening. In the Beis Hamikdash they would be said by the Leviim after the offering of the korban tomid shel shachar which corresponds to our shachris. The question is, why according to nusach ashkenaz is the shir shel yom said on shabbos and yom tov after MUSSAF. It would seem that nusach sefard is correct in saying the shir after shachris. [In fact Rav Shternbuch writes in his Tshuvos Vi-hanhagos that he changed his shul's minhag and instituted that it be said after shachris].
This question bothered the Acharonim and numerous answers have been offered. We will mention two:
1] Why do we say shir shel yom? Some say that it is זכר למקדש. Just as in the mikdash they said a special shirah every day we do so as well in order to fulfill the pasuk ונשלמה פרים שפתינו - our lips fill in for the bulls [especially if their flight from chicago was delayed and they couldn't make it to the game].
Others argue that this is not the reason. Our shmoneh esrei covers the obligation of ונשלמה פרים and the reason we read the shir is in order to fulfill the dictum of chazal [maseches sofrim] כל הקורא שירה בעונתה מעלה עליו הכתוב כאילו בנה מזבח והקריב עליו קרבן anybody who sings at the right time is considered as if he built a mizbeach and sacrificed a korban [see also Sanhedrin 101]. Saying the shir shel yom [which corresponds to what happened on that day during the seven days of creation] is singing the song at the right time.
According to this - the shir is not connected to the tomid shel shachar and there is thus no necessity to say it after shachris.
2] The second approach is that of Rav Hutner [Pachad Yitzchak Maamar ס"ז]. He Jewishly answers the question with a question. How come in shachris on Shabbos we don't mention the korban tomid shel shachar? [Ever notice that??]. In mussaf we do mention the korbanos that are brought on that day. The explanation is that shachris is not a fulfillment of ונשלמה פרים שפתינו but merely CORRESPONDS [is כנגד] to the tomid shel shachar, so we don''t have to mention the actual korban. Mussaf is actually a fulfillment of ונשלמה פרים so we have to mention the korban of the day.
Shockingly, in mussaf we mention not only the korban mussaf but the tomid as well. על שני תמידים כהלכתם. Why is that? It is definitely rooted in the fact that in the Torah when we are told to bring the mussaf we are told to bring the tomid as well, because mussaf is just an ADDITION [a mussaf] to the tomid shel shachar. It emerges that the shir shel yom that we say after mussaf is not related to mussaf in general but to the korban tomid that is mentioned in tefillas mussaf. The korban tomid is a fulfillment of ונשלמה פרים and it makes perfect sense to follow this with the shir shel yom.
According to this, explained the sefer Shiras Shmuel, we can understand why we say the words ישמחו במלכותך שומרי שבת in mussaf only [note that the sefer Abudraham already mentions ישמחו when explaining the tefilla of leil shabbos]. What does the simcha of shabbos have to do with mussaf? Explained Rav Moshe Soloveitchik [quoted by Rav Yosef Dov in his yahrtzeit shiur] that the pasuk says וביום שמחכתכם - on the day of your happiness, you should blow trumpets. The Sifree says that this day of happiness refers to Shabbos. Where do we find an obligation to have simcha on shabbos?? Simcha is usually associated with Yom Tov. The Ramban in the sefer hamitzvos teaches that Hallel on Rosh Chodesh was min hatorah based on the pasuk וביום שמחתכם ובמועדיכם ובראשי חדשיכם ותקעתם בחצוצרות - On Rosh Chodesh in the Beis Hamikdash they would blow trumpets. The gemara says that the main shira is not with trumpets but with song - עיקר שירה בפה. So just like we must blow trumpets on Rosh Chodesh we also must sing to Hashem בפה. However the gemara [Erchin 10] teaches that this doesn't apply outside the Beis Hamikdash [our hallel is only a minhag]. The same would them apply to Shabbos. The gemara there excludes Hallel on Shabbos outside the Beis Hamikdash because Shabbos is not a moed. But in the Beis Hamikdash it is still a mitzva to sing בפה and say Hallel. So the יום שמחתכם applies on Shabbos in the Beis Hamikdash where we were commanded to sing.
Since we established that mussaf is the tefilla that applies specifically to the korban in the Beis Hamikdash and is a fulfillment of ונשלמה פרים [as opposed to shachris which merely corresponds to the tomid shel shachar] - it makes sense that we should mention this idea of simcha ישמחו במלכותך שומרי שבת in the tefilla of mussaf, for as we saw, the simcha of Shabbos applied specifically in the Beis Hamkdash and the mussaf transports us back to the Beis Hamikdash.