The gemarah (Succah 44a-b) indicates that we beat (or according to Rashi, wave) the arava on Hoshana Rabba, what is the symbolic or theological reason behind doing this?
Teshuvas Hage'onim (Sharey Teshuva Siman 340) cites two reasons in the name of Rav Tzemach Goan:
The leaves of the arova look like lips. Hitting them on the ground hints to fact that we require atonement, and therefore fulfill the verse in Eicha (3:29) "יִתֵּ֤ן בֶּֽעָפָר֙ פִּ֔יהוּ אוּלַ֖י יֵ֥שׁ תִּקְוָֽה׃" (Let him put his mouth into the dust; there may yet be hope), meaning to say we have no further ability in our mouths to atone for ourselves.
Since during the days of Succos we fulfill many mitzvos, in this merit the accusations which the Satan made against us should be nullified. The Midrash compares the four species to various body parts; the arovah hints to the mouth. Hitting the arova on the ground, symbolizes our hope that the accusing mouth of the Satan shall be brought down to the ground. [A similar reason is brought by Tamey Haminhagim (Siman 819) in the name of the Tola'as Yaakov; to eradicate the judgement and to bring it down to the dust so it shall no longer exist in the world.]
The Levush (Siman 664 Sif 3) brings the custom of hitting the arova on the floor after hoshaynes, and then writes that all these sukkos customs are because of simcha.
This source cites the Eliyahu Zuta as saying that the falling away of the leaves that we beat symbolizes the falling away of "sparks of judgment" as the period of judgment comes to a conclusion.
Rabbi Shlomo Kluger writes (Chochmas Shlomo 664:3) that although there are certainly deep esoteric reasons behind the custom of beating the arovah, perhaps it could be explained according to the Midrash that the four species represent the four types of Jews: some have both smell and taste - Torah and good deeds - some have only one, and the arovah has neither smell nor taste representing the rosha who has neither Torah nor good deeds. The mitzvah of Lulav and Esrog is to bind all four species together signifying Jewish unity. However, Rabbi Shlomo Kluger writes, only in this world the r'shoim are unified with us; in the world to come there will a separate judgment for the wicked. Therefore, during the first six days of Sukkos - corresponding to the six thousand millennium of this world, we bind the arovah with the other species, and on the seventh day - Hoshana Rabba - corresponding to the world to come were we beat the arovah on the ground to subdue the wicked.
Rav Shimshon Raphael Hirsch writes in Chorev here (second paragraph) that after we have encircled the Bimah seven times we beat the Aravos as a means to remove any sense of Ga'avah (haughtiness / arrogance) from our hearts and the belief that our successes come from us alone. He says that the Aravah is symbolic of the 'chomer' lit. the material i.e. it is something that we receive from nature - it grows from the tree, wild and untreated, much like our natural abilities and talents. And with our human deficiencies, we wrongly assume that these talents are solely due to us. Therefore we specifically beat the Aravos to remind ourselves that everything is G-d given and beat out the natural arrogance that might express itself.