1) A CHILD'S OBLIGATION TO FULFILL THE MITZVAH OF LULAV
QUESTION: The Gemara rules that a Katan (minor) who knows how to shake the Lulav properly is obligated to fulfill the Mitzvah of Lulav.
The Gemara earlier, however, teaches that one fulfills the Mitzvah when he merely picks up the Lulav. Why, then, must a Katan be able to shake the Lulav in order to be obligated to fulfill the Mitzvah of Lulav? He should be obligated if he merely can lift it.
(a) The CHASAM SOFER (Chidushim) proposes that the requirement to shake the Lulav is a Mitzvah mid'Oraisa derived from a Halachah l'Moshe mi'Sinai. (This is implied by the Mishnah (29b, as explained by the Gemara on 32b) that says that the Lulav must extend at least one Tefach beyond the height of the Hadasim and Aravos "in order to shake it.") When the Gemara says that as soon as one picks up the Lulav he fulfills the Mitzvah mid'Oraisa, it means that he fulfills the Mitzvah only if he is fit to shake the Lulav. A young child or a sick person who is not physically able to shake the Lulav cannot fulfill the Mitzvah, because of the principle "Kol she'Eino Ra'uy l'Bilah, Bilah Me'akeves." (This approach of the Chasam Sofer is very novel and requires further elucidation.)
(b) RABEINU MANO'ACH (Hilchos Shevisas Asor 2:10) writes that the Rabanan instituted an obligation for a Katan to perform a Mitzvah for the purpose of Chinuch only when he performs that Mitzvah completely in accordance with the way an adult performs it. If the child does not perform the Mitzvah the way an adult performs it, then it is not considered Chinuch. (See also BRISKER RAV to Erchin 2a who reaches the same conclusion.)
Perhaps the reason for this is because a Katan is required to perform Mitzvos so that he learn how to fulfill the Mitzvos when he becomes an adult. Training him to perform only part of the details of the Mitzvah will not achieve this goal. On the contrary, he will learn how to perform the Mitzvah incorrectly, and "Shabeshta, Keivan d'Al, Al" -- once a child learns a mistake, the mistake stays with him (Pesachim 112a, Bava Basra 21a). Therefore, the Rabanan obligated a Katan to fulfill the Mitzvah of Lulav only when he is able to shake the Lulav properly. (See also Insights to Erchin 2:2.)
2) THE MITZVAH OF LULAV OUTSIDE THE BEIS HA'MIKDASH
OPINIONS: The Gemara earlier (29b) teaches a difference between the first day of Sukos and the other six days with regard to the Mitzvah of Lulav. On the first day of Sukos, the Mitzvah of Arba'as ha'Minim is mid'Oraisa. On the other six days, the Mitzvah is mid'Rabanan in all areas of the "Medinah" and "Gevulin" (outside of the Beis ha'Mikdash), as the Mishnah says earlier (41a). The Gemara here again raises this point in its discussion of when and where the Lulav may be held when Sukos occurs on Shabbos.
When the Mishnah and Gemara refer to the "Medinah" or the "Gevulin," the places where the Mitzvah of Lulav is mid'Oraisa only on the first day of Sukos, to what areas do they refer?
(a) RASHI (41a, DH b'Medinah) explains that "Medinah" refers to any part of Eretz Yisrael outside of the Beis ha'Mikdash, including the city of Yerushalayim. Only in the Beis ha'Mikdash itself is the Mitzvah mid'Oraisa all seven days of the festival. Although the words "Medinah" and "Gevulin" imply places that are far away, here it refers to any place outside of the Beis ha'Mikdash.
(b) The RAMBAM (in Perush ha'Mishnayos to Sukah 3:10, Ma'aser Sheni 3:4, Shekalim 1:3, and Rosh Hashanah 4:1) writes that "Medinah" refers to all of Eretz Yisrael outside of Yerushalayim. Yerushalayim itself, however, is included in "Mikdash," and thus in all of Yerushalayim the Mitzvah of Lulav applies mid'Oraisa for all seven days of Sukos. This also appears to be the opinion of RABEINU CHANANEL (44a, "b'Gevulin Zulas Yerushalayim").
The ARUCH LA'NER (41a, 43a; BIKUREI YAKOV OC 658) writes that nowadays if, during Sukos, one finds himself in the area of Yerushalayim which had Kedushas Yerushalayim, according to the Rambam he fulfills the Mitzvah d'Oraisa when he holds the Arba'as ha'Minim on all seven days. However, since the Mitzvah is mid'Oraisa according to the Rambam, one must be careful with regard to all of the Pesulim which normally apply only on the first day of Sukos (such as "Lachem" -- the requirement that one own the Lulav and not borrow it from someone else, and "Chaser" -- the requirement that the Lulav be complete), because those Pesulim apply all seven days mid'Oraisa in the "Mikdash" (according to Rashi and Tosfos on 29b; see Insights there).
The Acharonim discuss whether a person is obligated to go to Yerushalayim in order to fulfill the Mitzvah d'Oraisa with the Lulav (see MIKRA'EI KODESH, Sukos volume 2, p. 60 et seq.; MO'ADIM U'ZEMANIM, end of volume eight).
The OR SAME'ACH (in Chidushim to Sukah 42b) strongly opposes this view. He argues that nowadays when there is no Beis ha'Mikdash, Yerushalayim does not retain its Kedushah with regard to the Mitzvah of Lulav even according to the Rambam. The Rambam means that only when the Beis ha'Mikdash stands is there a Mitzvah d'Oraisa of Lulav in all parts of Yerushalayim. This is evident from the fact that Rebbi Yochanan ben Zakai saw the need to institute that the Arba'as ha'Minim be held for all seven days of Sukos as a "Zecher l'Mikdash." If the Mitzvah still applies mid'Oraisa for all seven days in Yerushalayim nowadays, why does it need to be commemorated with a "Zecher"? The Mitzvah needs no commemoration if it is still in effect mid'Oraisa in Yerushalayim for all seven days. (Another way of expressing this logic is that if the Mitzvah applies mid'Oraisa in Yerushalayim even when there is no Beis ha'Mikdash, then it clearly is not related to the Beis ha'Mikdash, and thus an enactment that it be observed in all places does not serve to commemorates the "Mikdash.") The Or Same'ach cites other proofs for his assertion.
It is important to note that in the Mishneh Torah, the Rambam makes no mention of the opinion he expresses in Perush ha'Mishnayos, that Yerushalayim is included in "Mikdash" with regard to the Mitzvah of Arba'as ha'Minim. Therefore, in practice, a person may be lenient and fulfill the Mitzvah with a Lulav that does not fulfill the requirement of "Lachem" during the last six days of Sukos in all parts of Yerushalayim. (See Insights to Rosh Hashanah 29:2