What are the connotations of "Vesamachta be'Chagecha" on Sukos?
Targum Yonasan: It refers to the Nisuch ha'Mayim ceremony and the accompanying flutes.
Chagigah, 8:1: It teaches us that one can bring Shalmei Simchah with all kinds of Basar - that a Yisrael can be Yotzei with Nedarim and Nedavos and with Ma'aser Beheimah, and a Kohen, with Chata'os and Ashamos, B'chor, Chazeh and Shok. 1
Mo'ed Katan, 8b: It implies "be'Chagecha", 've'Lo be'Ishtecha' - to prohibit getting married on the Yom-Tov, due to the principle 'Ein Me'arvin Simchah be'Simchah' (One may not mix one Simchah with another).
Mo'ed Katan, 14b: It teaches us that an Aveil does not apply Aveilus on Yom-Tov. 2
Yerushalmi Sukah, 4:5: The comparison of Simchah to Chagigah teaches us that the time of Shalmei Simchah coincides with that of Korban Chagigah. Consequently, since the Korban Chagigah is not eaten on the night of the fifteenth, 3 there is no obligation to eat Shalmei Simchah either. 4
Yerushalmi Chagigah, 1:1: It teaches us that a sick person is Patur from Re'iyas Panim ba'Azarah on Yom-Tov 5
Torah Temimah citing Chagigah (ibid.): But not with birds or with Menachos - which are not subject to Simchah. See Torah Temimah, note 60.
Torah Temimah (citing the Gemara): Even though Aveilus is also a Mitzvah, Aveilus de'Rabim overrides Aveilus de'Yachid.
See Torah Temimah, note 65.
See Torah Temimah, note 62.
See Torah Temimah, note 63, who discusses whether the obligation of Simchah nowadays is 'Oraysa or de'Rabbanan.
Why does the Torah need to specify "Atah u'Vincha u'Vitecha ... "?
Rashi (in Kidushin, 34a): To teach us that even women are Chayav in the Mitzvah of Simchah on Yom-Tov. 1
Pesachim, 109a: To teach us that one is obligated to make one's children and other members of one's family happy - the men with wine, the women with nice clothes. 2
With what will the Ger, the Yasom and Almanah rejoice?
Seforno: With the Peret and Olelos 1 and gifts [for the poor] from the fruit of the trees.
Of the vines, corresponding to the Leket of grain. Refer also to 16:11:3:2**.