Dear Rov, I have some questions that have bothered me a long time.
Q) Is there a mitzva of simcha on Rosh Hashana, as in every other Yom Tov, so a vegetarian should make an effort to eat meat?

A. There are conflicting sources and opinions as to the mitzva of rejoicing on Rosh Hashana. Zohar (Emor 95) mentions that there is no “chedveza” or rejoice on the days of Rosh Hashana and Yom Kipur, as they are days of judgment. It would also seem so from the fact that we don’t refer to Rosh Hashana as a Chag in the Tefilos, as opposed to Chag Happesach, Hag Hassukot etc. Also Shulchan Aruch (O.H. 582: 8) rules that we don’t say; moadim lesimcha chagim uzmanim lesasson, on them, There are also no Shalmei Simcha sacrifices as in all other Yomim Tovim. Maharil (end of H. Yom Tov p.25) outright rules that there is no mitzvas simcha and you are allowed to fast on Rosh Hashana. Similarly, Turei Even (Chagiga 8b) opines that there is no mitzvas simcha and you are therefore allowed to mix in on Rosh Hashana other simches.
However, the pasuk (Tehilim 81: 4) does address Rosh Hashana as a chag as it says; Bakesse Leyom Hageinu. And Nechemia (8: 9-10) does call on Rosh Hashana to the people that; “This day is holy to Hashem; neither mourn nor weep… Go, eat fat foods and drink sweet drinks and send portions to whoever has nothing prepared, for the day is holy to our G-d, and do not be sad, for the joy of Hashem is your strength.”
Sha’agas Aryeh (20) quotes other sources that prove that there is indeed a Mitzva of simcha. That is the opinion of many Rishonim, including the Yereim, the Ran and Rif on Beitza. Shulchan Aruch (O.H. 597: 1) also rules that it is a mitzva to rejoice, as does the Mishna Berura (ibid. 1). Poskim quote also the all Yomim Tovim, including Rosh Hashana are compared to each other. (by hekesh).
Moadim Uzemanim (6: 12, & 7: 114) tries to bridge the conflicting opinions, following the view of the Rambam (H. Chanuka 3: 6) on why we don’t recite hallel on Rosh Hashana, he opines that since they are days of teshuva and concern, there should be joy but not excessive as in other Yomim Tovim.
Horav Shlomo Miller’s Shlit’a opinion is that the simcha should be seen as “gilu bereada” (Tehilim 2: 11) or rejoice, while trembling. Therefore, the Rov maintains that one should eat meat as he does on the other Yomim Tovim.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a