The Senate then began hearing the views of House managers and Trump's lawyers regarding a resolution proposed by Senate Majority Leader Mitch McConnell, R, that seeks a swift trial.
A simple majority of senators - 51 votes - is needed to approve the resolution and other motions in the Republican-controlled chamber.
Democrats, including Senate Minority Leader Chuck Schumer, have panned the proposal, arguing that it is part of an effort to "cover up" Trump's dealings. In public appearances, the House impeachment managers urged the Senate to reject McConnell's rules.
Before the trial, McConnell defended his resolution as "straightforward" and "imminently fair" and said he would resist any amendments. During today's trial, Schumer moved to amend the rule package to allow for witnesses and documents at the front end of the trial.
The House last month approved two articles of impeachment against Trump: abuse of power and obstruction of Congress. The impeachment charges centre on the allegation that Trump withheld military aid and a White House meeting to pressure Ukraine to investigate his political rivals, including former Vice-President Joe Biden and his son Hunter Biden.
McConnell's resolution initially offered each side 24 hours to make its opening arguments, starting tomorrow, but compressed into two session days. But today, the resolution was changed to relax the timetable for arguments and ease the admission of evidence, according to two Republican aides.
The 24 hours of arguments from impeachment managers and the White House will now take place over three days instead of two, and evidence will be automatically admitted unless someone objects, the aides said.
Senator Susan Collins, a pivotal swing Republican vote on procedural matters, had raised concerns privately about both the evidence provisions and the truncated number of days that both the House impeachment managers and the president's lawyers would have to argue their cases.
In a brief opening statement, White House Pat Cipollone cast the impeachment trial as "a fair process" and urged senators to support McConnell's resolution.
"We believe that once you hear those initial presentations, the only conclusion will be that the president has done absolutely nothing wrong and that these articles of impeachment do not begin to approach the standard required by the Constitution, and in fact they themselves will establish nothing beyond those articles," Cipollone said.
Congressman Adam Schiff, D, the lead House impeachment manager, argued against McConnell's resolution, saying it needs to be changed to ensure that House managers can obtain documents relevant to Trump's conduct toward Ukraine as well as relevant witnesses.
In his remarks, Schiff said the vote on the resolution would be even more important than the vote senators will take on Trump's guilt.
"I believe the most important decision in this case is the one you'll make today," he said. "Will the president and the American people get a fair trial?"
Under McConnell's resolution, decisions about documents and witnesses would be put off until after House managers and Trump's lawyers make opening arguments and senators have an opportunity to ask questions of both sides.
Schiff said that order makes no sense.
Jay Sekulow, an outside lawyer for Trump, began his floor remarks with several inaccurate statements.
Sekulow incorrectly claimed that the report issued last year by former special counsel Robert Mueller found "no obstruction" - when in actuality, Mueller declined to weigh in on the question of whether Trump obstructed justice, citing long-standing Justice Department precedent.
Sekulow also claimed that the White House was not allowed to have a counsel present during the impeachment hearings spearheaded by the House Judiciary Committee. In fact, the White House was allowed to participate but declined to do so.
Throughout today's hearing, most senators were taking notes. Several also appear to have brought reference materials with them, most in three-ring binders. Reporters can only observe about two-thirds of the senators from any part of the press gallery.
After the opening remarks, the trial moved to two hours of debate on Schumer's amendment, which would have the Senate subpoena White House documents related to the charges against Trump.
Schiff took the floor first. He began by rebutting some of the claims made by the President's lawyers, including Cipollone's assertion that Republicans were not allowed in the closed-door impeachment depositions.
"Every Republican on the three investigative committees" was allowed to participate, Schiff said, calling Cippolone "mistaken" on the matter.
Earlier, in a letter to Cipollone, the House impeachment managers suggested he may have a conflict that should keep him from representing Trump in the Senate trial. The managers are Schiff; Jerrold Nadler; Zoe Lofgren; Hakeem Jeffries; Val Demings; Jason Crow; and Sylvia Garcia.
The House managers noted several instances in which it appeared that Cipollone was aware of Trump's conduct toward Ukraine for which the President has been impeached.
"In light of your extensive knowledge of these key events, your personal representation of President Trump threatens to undermine the integrity of the pending trial," the managers said. "You may be a material witness to the charges against President Trump even though you are also his advocate."
The White House dismissed those concerns."The Democrats are an utter joke - they have no case, and this latest political stunt proves it," said White House spokesman Hogan Gidley.
