The form of prayer required on Rosh Hashanah
One should pray with profound bitterness, having recognition that one is approaching the King with “torn garments”, and thus begging G-d to have mercy and accept him as his servant. Nevertheless, when one accepts the yolk of heaven it must also be done with joy and desire.
What to Daven for:
The Rebbe Rashab stated: “On Rosh Hashanah one is to pray with great fervor, accompanied by tears of mercy, that one be successful in amending his character and acting in a Chassidic method.” The Zohar states that those that pray on [Rosh Hashanah and] Yom Kippur merely for physical matters are similar to a dog that screams “give give” for their food. This is the attribute of the Eirev Rav and the gentiles that they are only able to serve G-d for reward and selfish motivations. Rather a Jew is to arouse himself in Teshuvah out of love for the Shechina and desire for it to leave the exile.
Crying during prayer:
The Arizal would cry profusely during the prayers of Rosh Hashanah and Yom Kippur, and so was the custom of the Chabad Rabbeim, and the city of Lubvatich. The Arizal said that if one is not drawn to cry during these days [of Rosh Hashanah and Yom Kippur] it is a sign that his soul is incomplete, G-d forbid. This cry that is required is one of passionate longing for G-d. One is however not to cry on Rosh Hashanah due to a loss in physical or even spiritual matters. This however only applies to Tzaddikim Gemurim, however those which still contain evil in their hearts, certainly are to cry over the evil that is in them which prevents them from attaching to G-d. If one is strong hearted and is unable to bring himself to tear than he is at the very least to make a crying sound, and Hashem will hear the attempted cry of his heart.
Rosh Hashanah that falls on Shabbos: One is to cry on Rosh Hashanah even if it coincides with Shabbos.
Crying without reason: One who is suddenly overcome with an outburst of tears on Rosh Hashanah is a sign that at this moment his soul is being judged before Hashem.
The affect of tears: One who cries out of Teshuvah on Rosh Hashanah has the ability to tear the evil decree and write him into the book of Tzaddikim. The Zohar states that crying during prayer is a great Segula for one’s prayer to be accepted above.
Wiping the tears over one’s face: It is proper to wipe the tears over one’s face as the sins of a person are written on his forehead [and are hence cleansed by these tears].
May one pray for physical matters on Rosh Hashanah?
Yes. One may make requests of physical and spiritual matters on Rosh Hashanah even if it coincides with Shabbos.
 Sefer Hamamarim Kuntreisim 2 p. 323; See also Sefer Hamamarim 1943 p. 42
 Yahel Or p. 289
 Sefer Hasichos 1944 p. 11
 Reishis Chochma Shaar Hateshuvah chapter 1
 Shaar Hakavanos 90a “My teacher the Arizal cries profusely during the prayers of R”H even though it is a Yom Tov and certainly during the prayers of Yom Kippur”; Peri Eitz Chaim 25/5; brought in Beir Heiytiv 584/3; Mateh Efraim 582/28; Kaf Hachaim 582/60
Other Opinions: In Maaseh Rav 207 it is written of the Gr”a that one is not to cry on Rosh Hashanah, as the verse in Nechemia [8/9] states “Al Yivku” “Do not mourn or cry. Go eat fatty meats and sweet beverages. Don’t be sad as joy of G-d is your rock.” However see the following footnotes for those that interpret the Gr”a to refer only to a cry of pain.
 Toras Menachem 1951 Vol. 2 p. 4; Sefer Hasichos 1947 1st night R”H “The Alter Rebbe once said that the Avoda on the night of R”H must be accompanied with joy. The Tzemach Tzedek and Reb Nachum then asked the Alter Rebbe “Why is it then that you cry” He replied that it is a cry of joy”; Sefer Hasichos 1941 p. 27 “The Rebbe Rashab’s holy face was soaked with tears”; Hatamim 2 p. 131-132 “The Rebbe Rashab Davens and cries. There isn’t one word he recites that isn’t immersed in a river of tears
 Sefer Hasichos 1944 p. 5 “Even the citizens of the town of Lubavitch, men, women and children on the two days of Rosh Hashanah were filled with cries of repentance.”
 Shaar Hakavanos 90a; Peri Eitz Chaim 25/5; brought in Beir Heiytiv 584/3; Mateh Efraim 582/28; Kaf Hachaim 582/60
The reason: The reason for this is because on Rosh Hashanah one is to naturally feel the attachment of his soul to its root in G-d, and this feeling is so intense that the body sheds tears out of inability to contain it. Hence one who does not cry shows that his soul is no longer sensitive to its attachment to G-d. Hence the cry which is required is one of passionate longing for G-d. One is however not to cry on Rosh Hashanah due to a loss in physical or even spiritual matters, as explained in sources in next footnote. One who is not brought to tears on Rosh Hashanah shows that he has a hard heart and a cruel personality. [Alef Hamagen 582/47; Piskeiy Teshuvos 584/2]
 Likkutei Torah Ki Seitzei “Vehisira Es Simlas Shivya” [p.72]; “Ubachsa Es Aviha” [p.74] [“This cry is not due to Marah Shechorah” “Is similar to the cry of Rebbe Akiva when he said Shir Hashirim” “Is level of Teshuvah Ilaah”]; See Sefer Hasichos 1947 1st night R”H brought earlier “The Tzemach Tzedek and Reb Nachum asked the Alter Rebbe why he cries and he replied that it is a cry of joy; Shaar Yissachar Tishrei “The cry must be a result of Dveikus in prayer”; The Gr”a in Maaseh Rav 207 rules that one is not to cry on Rosh Hashanah, and it is brought in the name of Rav Chaim Volozhin that the intent of this statement is that one is not to cry due to physical matters or matters of negativity but rather it is to be cries of joy, as requires the Arizal. Thus those Chazanim that bring the congregation to cry over the words “On Rosh Hashanah we are written..” are doing an improper act. [See Piskeiy Teshuvos 584/2 footnote 15; Teshuvos Vehanhagos 268 explains this to be the opinion of Gr”a]
 Likkutei Sichos 19 Parshas Haazinu
 Alef Hamagen 582/45 in name of Vavei Amudim in name of Shalah; Shiyurei Bracha [source not found]; Sdei Chemed Mareches Rosh Hashanah 2/26 in name of Chida
 M”E ibid; implication of Shaar Hakavanos ibid, brought in Kaf Hachaim ibid
 Shaar Hakavanos 90a; brought in Beir Heiytiv 584/3; Mateh Efraim 582/28; Kaf Hachaim 582/60
 Alef Hamagen 582/44
 Totzos Chaim; Reishis Chochmah Shaar Hayirah 9; Kaf Hachaim 582/60
 Mateh Efraim 584/25 [regarding an ill person]; Az Nidbaru 14/21; Piskeiy Teshuvos 582/7
 Although asking for one’s physical needs is ordinarily forbidden on Shabbos and Yom Tov [See Admur 294/1; 299/15; 188/4; 584/5] nevertheless it is permitted to do so on Rosh Hashanah being that this day was established to be a day of judgment which requires prayer and supplication. [ibid] Vetzaruch Iyun from the ruling in 584/5 regarding Avinu Malkeinu, that it is omitted on Rosh Hashanah due to Bakashas Tzerachav! Some suggest that only a set prayer of request was negated on Shabbos Rosh Hashanah while a private request is not only allowed but is motivated to be done. [Piskeiy Teshuvos 582 footnote 21]
Where in Shmoneh Esrei are the requests to be made? The requests are to be made after the first Yehiyu Leratzon but prior to reciting the second Yehiyu Leratzon. [See Admur 122/1; M”B 12/3]