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Rav Yehoshua Grunwald
(ביום השמיני עצרת תהיה לכם כל מלאכת עבודה לא תעשו” (במדבר כט:לה”
The Midrash states that when Hashem asked Klal Yisrael to remain with Him for the extra day of Shemini Atzeres, they responded, “nagila vnismecha bo.” The Midrash explains that bo means rejoicing with Hashem, rather than the day itself, based on the pasuk “nagila vnismecha bach [you-Hashem]”: Bach betorascha, bach beyeshuascha.
Rabbeinu Yona explains that our primary simcha on Yom Tov is for the miracles performed for us specifically. We can add that commemorating the open miracles performed for our ancestors linked to each Yom Tov enables us to recognize the hidden miracles in our own lives, the primary subject of our thanks on Yom Tov. Rabbeinu Yona also states that when we recognize his miracles, we can actually see Him in a certain sense (bach beyeshuascha), and the same is true with studying Torah (bach betorascha).
Why does the Midrash discuss the nature of simcha on Yom Tov concerning Shemini Atzeres specifically and not other Yomim Tovim? The answer is based on the Sefer HaChinuch, which explains that the mitzvos of Sukkos are designed to increase our simcha, but on Shemini Atzeres they are not necessary, as our remaining with Hashem and experiencing a special closeness with Him suffices to provide us with this simcha. Therefore, in contrast to other Yomim Tovim, on Shemini Atzeres we say Hallel without commemorating any miracles, since the simcha is solely about closeness to Hashem: Bach bishuascha bach betorascha.