[Avodah] Shmini Atzeret - why Sukkah YES and Lulav NO?

[Marty Bluke]

RYBS has a nice explanation of this (printed in the shiurim on Succah
47a). He explains that on Shemini Atzeres chazal had a problem with
Yom Tov Sheni. On all other YT Sheni's they imposed a higher kedusha
d'rabbanan of Yom Tov on either Chol Hamoed or on a yom chol. The
Kedushas hayom of Yom Tov is what is mechayev you in the mitzvos. On
Shemini Atzeres however, there is already a kedushas hayom of Yom Tov
d'oraysa of Shemini Atzeres and therefore the chahamim could not
impose a lower kedusha d'rabbanan of chol hamoed succos. Therefore, he
says that Shemini Atzeres has no kedushaa hayom of succos at all, and
therefore there really is no chiyuv for any of the mitzvos of succus.
However, because of the idea of YT Sheni shel galuyus the chachamim
were mechayev you to sit in the succah without a beracha as a separate
chiyuv from the regular mitzva of succah.

He is madayek this both from Rashi in Succah (47a). Rashi explaining
why there is no beracha states: "d'shmini hu v'ayn shem succos alav".

The Rambam in perek 6 halacha 13 writes:
"... u'bayom hashmini shehu yom tov rishon shel shmini atzeres yoshvin
ba v'ayn mevarchim" The phrase "shehu yom tov rishon shel shmini
atzeres" seems to be superflous, why did the Rambam need to say it.
>From this RYBS is medayek that there is no kedushas hayom of succos at
all.

Based on this we understand why there is no lulav as well.