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Marbim Besimcha: 
One is to increase in joy throughout the month of Adar. [Hence one is to rejoice himself, his wife and children during this month, similar to the Mitzvah of rejoicing them on Yom Tov. Included in this Mitzvah is increasing in all matters that give one joy. However foremost, one is to add in spiritual matters of joy such as to increase in learning Torah, in particular the inner dimension of Torah, and the scrupulous observance of Mitzvos.]
Does one increase in Simcha on the first day of Rosh Chodesh which is the 30th of Shevat?
Yes. However the main increase of joy is on the second day of Adar, which is after a full day of the month of Adar has passed.
During a leap year does the Mitzvah to increase in joy apply to both months of Adar?
See next Chapter in First Halacha!
Dancing every night of Adar:
Many Chabad Yeshivas have the custom of dancing each night of the month of Adar, in light of the command of the Sages to increase in joy during this month.
Increasing in Simcha every day, even after Purim:
One is to add in Simcha on every day of the month of Adar. This includes even the days after Purim in which one is to increase his joy from one day to the next. This is especially true in light of the fact that one is coming closer to the redemption of Pesach, and hence is approximating one redemption to the next.
Sparks of Chassidus
What does the word Adar stand for?
The Alef represents Alufo Shel Olam, which refers to Hashem. The word Dar means to live. In the month of Adar we are able to accomplish that Hashem dwell below in this world and fulfill Dirah Betachtonim.
 Magen Avraham 686/5 based on Gemara Taanis 29a
Why was this law omitted by the Michaber? See Chasam Sofer 160; Nimukeiy Orach Chaim 687 as for why this law was not mentioned by the Rambam or Michaber. The Rebbe explains that perhaps the reason the Rambam omitted this law is because he rules one must be in a state of joy throughout the entire year. This is in contrast to the opinion of the Talmud which states only by Adar that one is to increase in joy, hence implying that it is not proper to have extra joy during other months of the year. [Igros Kodesh 27/114] Alternatively the reason is because there is no action involved in this Mitzvah to increase of joy, as it is merely an addition in one’s feeling, and hence the Rambam did not feel a need to bring it. [Hisvadyos 1991 2 p. 300 Edited]
The comparison to the month of Av: The Gemara ibid states that “just like as Av enters one diminishes in Simcha, similarly when Adar enters one must increase in Simcha.” Regarding the reason for this comparison-see Shaareiy Hamoadim Adar p. 29. In Likkutei Sichos 16/345 the Rebbe explains that just like the entire month of Av is instinctively a month of sadness and mourning and due to this nature this month was allocated to host the bad occurrences, similarly the month of Adar is intrinsically meritorious and it is for this reason that it merited to host the miracles of Purim.
 The M”A ibid writes based on the Gemara ibid “When Adar enters one is to increase in joy”. The Ashel Avraham Butchach 686 Mahadurah Tinyana writes this Mitzvah applies throughout the month of Adar. See Likkutei Sichos 16/345 brought in previous footnote.
The reason: Rashi [on Taanis 29a] explains that the reason for increasing in joy in Adar is due to the miracles that occurred on Purim and Pesach. The Rebbe [Likkutei Sichos 16/345] explains Rashi to mean that the entire month is a meritorious month for the Jewish people due to the fact that Moshe Rabbeinu was born on this month, and it is due to his merit that we merited the miracles of Purim and Pesach. This is why Rashi also mentioned Pesach in his explanation. Shieilas Yaavetz 2/88 explains the reason for mentioning Pesach is because this shows that the miracles continued for two months and hence we increase in joy in Adar and not Chanukah, as by Chanukah the miracles were for only one month.
The month of Nissan: Some Poskim write one is also to increase in joy in the month of Nissan. [Elya Raba 685/8; Chasam Sofer 160 based on Rashi Taanis 29a; See Likkutei Sichos 16 p. 344 brought in previous footnote]
 Sefer Hasichos 1992 p. 391
 Nimukeiy Orach Chaim 687; Likkutei Sichos 4/1274 [Such as planting trees of Kings for shade and the like-ibid]; Likkutei Sichos 16/345 [Its not specifically adding in feasts and merry making, but any type of Simcha]
 Sefer Hasichos 1992 p. 391
 Shaareiy Moadim Adar 1 p. 20
Shabbos Mevarchim Adar: On a number of occasions the Rebbe stated that one is to begin increasing in joy from Shabbos Mevarchim Adar. [Hisvadyos 1991 p. 300; See Oatzer Minhagei Chabad 17]
 Likkutei Sichos 4 p. 1274
 As if one were to have the same amount of joy each day then it would become habitual and common, thus losing its enthusiasm. This is similar to the requirement to have “Panim Chadashos” by each Sheva Brachos as otherwise the joy begins to become stale. [ibid]
 Sichas Shabbos Para 5731
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