What do we learn from "Hayu Tzareha l'Rosh"?
Sanhedrin 104b: Whatever nation afflicts Yisrael becomes the dominant power 1 (Tosfos - beforehand, lest people say that Yisrael fell to a lowly nation). Ri Kara - "Hen Eretz Kasdim Zeh ha'Am Lo Hayah" (Yeshayah 23:13); in order that it will destroy Yerushalayim, it became the head (Chagigah 13b). When Yisrael did Hashem's will, "u'Nsancha Hashem l'Rosh v'Lo l'Zanav" (Devarim 28:13). After they sinned, their enemies were at the head (Pesichah to Eichah Rabah 11).
Rav Sadya Gaon: They were heads and authorities.
Ibn Ezra: Each of its afflicters became a head 2 . This is like "Bnos Tza'adah Alei Shor" (Bereishis 49:22).
Sanhedrin 104b: "Ki Lo Mu'af la'Asher Mutzak Lah" (Yeshayah 8:23) - whatever nation afflicts Yisrael does not weary.
Magihah (in Toras Chayim): He explains why Hayu is plural, and l'Rosh is singular.
Why does it mention "Tzareha" and "Oyveha"?
R. Avigdor (from Eichah Rabah 1:31): Tzareha refers to Nebuchadnetzar, about whom it says "Ant Hu Reishah Di Dahava" (Daniel 2:38). "Oyveha" is Nevuzaradan, who destroyed Yerushalayim.
Palgei Mayim: This laments their great affliction. Anyone who afflicts them becomes a head. Enemies who did not afflict them have great serenity. The more than one afflicts them, his reward is greater. It seems that Hashem despised Yisrael 1 !
What is "Shalu"?
Rav Sadya Gaon, Rashi: They had Shalvah (serenity). Ri Kara - this is like "Shalev Hayisi va'Yfarpereni" (Iyov 16:12). Rashbam - The Vov is from the root, even though it has a Shuruk.
What is the meaning of "Hogah"?
Rav Sadya Gaon: He pushed them from in front of Him.
Ibn Ezra: This is from Yagon (He made them sad).
Ri Kara It is an expression of breakage, like "Ki Im Hogah v'Richem k'Rov Chasadav; Ki Lo Inah mi'Libo va'Yageh Bnei Ish" (3:32-33).
Palgei Mayim: It is removal, like "Hago Sigim mi'Kasef" (Mishlei 25:4). Hashem removed His Hashgachah from them, due to "Rov Pesha'eha."
What is the meaning of "Halchu Shvi"?
Ibn Ezra: It is like Halchu b'Shevi (they were taken captive). The prefix Beis is omitted, like "ha'Nimtza Veis Hashem" (Melachim II, 12:11).
Why does it specify the infants?
Ibn Ezra: It was due to Pesha'eha (the adults' sins).
Ri Kara: It is not normal for infants to be taken captive.
Palgei Mayim: This proves that Hashem removed His Hashgachah. Infants, who did not sin, were punished!