The Mishna in Sukkah (37b) commences with Beis Hillel's opinion that one must wave the lulav when reciting the pesukim of "Hodu Lashem, ki tov" ("Give thanks to Hashem, for He is good") and "Ana Hashem hoshi'a na" ("Please, Hashem - please save") during Hallel. The Medrash explains that after being judged on Rosh Hashanah, B'nei Yisroel emerge favorably and are thereupon commanded to perform the mitzvah of lulav, rejoicing in their meritorious verdict by waving their lulavim (as celebratory banners), like a jubilant person who has just been vindicated by a judge. The Medrash connects the pasuk of "Az yeranenu atzei ha-ya'ar" - "then shall the trees of the forest happily sing" (Divrei Ha-Yamim I 16:33) - symbolized by the lulav - with the motifs of thanksgiving and prayer for salvation featured in the immediate two subsequent pesukim (34-35), which are almost identical with the two above-cited pesukim of thanksgiving and prayer for salvation in Hallel, as an allusion to the waving of the lulav when reciting "Hodu Lashem, ki tov" and "Ana Hashem hoshi'a na" during Hallel. (Please see Rosh ibid. s. 26 for elaboration.)
It is perplexing that the Medrash relates themes of simcha and thanksgiving, reflected by the lulav's imagery of victorious emergence from judgment, with pesukim of pleas for salvation ("Ana Hashem hoshi'a na", etc.), the latter of which naturally evoke notions of trouble and anxiety. Does the lulav characterize cheerful celebration or supplications for help and salvation? How can the lulav characterize both of these concepts, which are quite contradictory?
The Rambam (Hilchos Lulav 7:23) rules according to the opinion in the Gemara (Sukkah 43b) that every day in the Beis Ha-Mikdash, the Kohanim would march around the Mizbe'ach (Altar) with their lulavim and recite "Ana Hashem hoshi'a na...". Rav Yosef Dov Ha-Levi Soloveitchik zt"l maintained (Reshimos Shiurim - Sukkah ibid.) that this practice was a fulfillment of the mitzvah of "u'semachtem lifnei Hashem Elokeichem shivas yamim" - "and you shall rejoice before Hashem your God for seven days" (Vayikra 23:40), stated in reference to Sukkos and in particular in reference to the mitzvah of lulav in the Beis Ha-Mikdash according to the Rambam, based on the Yerushalmi (Sukkah 3:11). The question arises here again - how can a mitzvah manifesting simcha, of "u'semachtem lifnei Hashem Elokeichem shivas yamim", be performed while reciting verses begging for salvation? Is this not a true paradox?
Although being in the presence of Hashem on Sukkos, as expressed by the pasuk of "u'semachtem lifnei Hashem Elokeichem shivas yamim", imbues one with a palpable sense of simcha, the simcha is not merely a function of being in the proximity of the Shechinah, so to say. Rather, the simcha of Sukkos is engendered by being in Hashem's presence with a sense of security, knowing that one is in Hashem's hands, relying on Him for salvation and all else; this endows one with a sense of joy, serenity, contentment and peace. It is this simcha of "Ana Hashem hoshi'a na", of being dependent on Hashem and solely in His omnipotent care, that brings the authentic inner joy of Sukkos. It is not a plea to Hashem of anxiety and trouble, but of closeness and love, like an infant in the caring arms of its mother.
The Gemara (Sukkah 37b) addresses the symbolism of waving the lulav: Rabbi Yochanan said, "One moves (the lulav) back and forth for the sake of Him to Whom the four winds belong; one lifts and lowers (it) for the sake of Him to Whom are the heavens and the earth... Rabbi Chama bar Ukva said in the name of Rabbi Yose bar Rabbi Chanina, "One brings (the lulav) back and forth to prevent bad winds; one lifts and lowers (it in supplication) to prevent bad precipitation." The latter opinion clearly does not maintain that waving the lulav exhibits worry and apprehension regarding bad winds and precipitation; on the contrary, the lulav is identified with simcha. Rather, the supplication to Hashem to prevent bad winds and precipitation is a reflection of being in Hashem's tender care, such that we lovingly turn to Him for protection and security.
This feeling of closeness to Hashem while experiencing exclusive reliance on Him for safekeeping and salvation, is the true sensation of simcha effected by the lulav.
Wishing everyone a good Yom Tov.