The constitution does not mandate a particular date for the inauguration, nor the end of the terms of elected officials, and by extension, the beginning of new terms. Therefore, it was a subject of legislation. In early days of the republic, when agrarian society was most prominent, November of December was considered a good time for elections, as harvests and the autumn slaughter of livestock was finished, but severe winter weather had not set in. In 1845, the date for elections was set at the first Tuesday after the first Monday in November. This assures an election as early as possible in November, but never on November 1st, which is a religious holiday in many christian churches. Holding the inauguration on March 4th allowed time for the states to certify their election results in the days before electronic communications, and for the electors to assemble to certify the results. It’s a little more complicated than that, because if no president has been chosen, then the House chooses the President. This happened in 1824 when Andrew Jackson and John Quincy Adams both polled 89 electoral votes. The House chose Adams, largely out of spite against Jackson. They were a lame duck Congress, and Jackson had many enemies in that Congress. By and large, setting Inauguration day on March 4th was a concession to the seasons and poor roads.
In 1932, an amendment to the constitution was proposed, in large measure to propose a solution to the problem of a lame duck House possibly selecting the President. It was approved pretty rapidly, and became the 20th Amendment. It sets the date for the new session of Congress for January 3rd, and the presidential inauguration for January 20th. This eliminates the lame duck House problem. Previously, the new Congress was seated on March 4th also, assuring that a lame duck House would choose the President in a contested election, or in the case of the death of the winning candidate. The 20th amendment is known as the Lame Duck Amendment.
Here is the text of the amendment:
Amendment XX
Section 1.
The terms of the President and Vice President shall end at noon on the 20th day of January, and the terms of Senators and Representatives at noon on the 3d day of January, of the years in which such terms would have ended if this article had not been ratified; and the terms of their successors shall then begin.
Section 2.
The Congress shall assemble at least once in every year, and such meeting shall begin at noon on the 3d day of January, unless they shall by law appoint a different day.
Section 3.
If, at the time fixed for the beginning of the term of the President, the President elect shall have died, the Vice President elect shall become President. If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified; and the Congress may by law provide for the case wherein neither a President elect nor a Vice President elect shall have qualified, declaring who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified.
Section 4.
The Congress may by law provide for the case of the death of any of the persons from whom the House of Representatives may choose a President whenever the right of choice shall have devolved upon them, and for the case of the death of any of the persons from whom the Senate may choose a Vice President whenever the right of choice shall have devolved upon them.
Section 5.
Sections 1 and 2 shall take effect on the 15th day of October following the ratification of this article.
Section 6.
This article shall be inoperative unless it shall have been ratified as an amendment to the Constitution by the legislatures of three-fourths of the several states within seven years from the date of its submission.