The Mishna in Sukkah (37b) commences with Beit Hillel's opinion that one must wave the lulav when reciting the verses, in the Hallel prayer said on Sukkot, of "Hodu Lashem, ki tov" ("Give thanks to Hashem, for He is good") and "Ana Hashem hoshi'a na" ("Please, Hashem - please save").
The Midrash explains that after being judged on Rosh Hashanah, the Children of Israel 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 Midrash connects the verse of "Az yeranenu atzei ha-ya'ar" - "then shall the trees of the forest happily sing" (Chronicles I 16:33) - symbolized by the lulav - with the motifs of thanksgiving and prayer for salvation featured in the immediate two subsequent verses (34-35), which are almost identical with the two above-cited verses 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 Tos. ibid. d.h. "B'hodu" for elaboration.)
It is perplexing that the Midrash relates themes of simcha and thanksgiving, reflected by the lulav's imagery of victorious emergence from judgment, with verses 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 (Laws of Lulav 7:23) rules according to the opinion in the Talmud (Sukkah 43b) that every day in the Tenple, 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 (Reshimot Shiurim - Sukkah ibid.) that this practice was a fulfillment of the mitzvah of "u'semachtem lifnei Hashem" - "and you shall rejoice before Hashem" (Lev. 23:40), stated in reference to Sukkot and in particular in reference to the mitzvah of lulav in the Tenple according to the Rambam, based on the Jerusalem Talmud (Sukkah 3:11).
The question arises here again - how can a mitzvah manifesting simcha, of "u'semachtem lifnei Hashem", be performed while reciting verses begging for salvation? Is this not a true paradox?
Although being in the presence of Hashem, as expressed by the verses of "u'semachtem lifnei Hashem" and of "lifnei Hashem tit'haru" - "you shall become purified before Hashem" (Vayikra 16:30) - imbues one with a palpable sense of simcha, the simcha is not merely a function of being in the proximity of the Shechinah, G-d's Presence, so to say. Rather, the simcha 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 authentic inner joy. 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.
This feeling of closeness to Hashem while experiencing exclusive reliance on Him for protection, security and salvation, is the true sensation of simcha effected by the lulav.
Wishing all a good Yom Tov.