Trump seemingly played down the importance of the exercises during a news conference after his summit in Singapore with the North Korean leader, in which Kim broadly agreed to end his nuclear weapons program without providing virtually any specifics on how he would do so.
The official start of the summit came when the two world leaders shook hands shortly after 9 a.m. local time.
Earlier, before a row of alternating USA and North Korean flags, the leaders shared a warm 13-second handshake at a Singapore island resort, creating an indelible image of two unorthodox leaders as they opened a conversation that could determine historic peace or raise the specter of a growing nuclear threat.
The media spotted Trump and Kim approaching the limousine, adorned with an American flag.
The first meeting that the pair had on their schedule was one-on-one, accompanied only by translators.
Kim said that "it was not an easy path here", but went on to say that "we've overcome everything and come to this place".
The two leaders will likely have to conduct reciprocal visits to each other's capitals over coming years before hatching out a final deal.
Trump did not refer to the document as a treaty or agreement.
President Donald Trump and North Korea's Kim Jong Un concluded an extraordinary summit Tuesday with lofty promises by the American president to take care of a "very unsafe problem" and Kim forecasting "major change for the world".
The letter, which was publicly disclosed after Trump signed it with a flourish and held it up to the TV cameras, contained the grand language typical of Trump.
President Trump says North Korea's Kim Jong Un has made an "unwavering commitment" to complete denuclearization after an historic meeting in Singapore. Read it in full below.
"Trump seemed to be very aware of this, that he needed to up the stakes and be seen that he is the leader".
"I do trust him, yeah", Trump said. "It's possible. We're dealing at a high level, a lot of things can change a lot of things are possible". There are even discussions of a Nobel peace prize for Trump and the Kim.
Kim was accompanied at the meeting by vice-chairman of the Central Committee of the Workers' Party of Korea (WPK) Kim Yong Chol, chairman of the Foreign Affairs Committee of the WPK Ri Su Yong, and Colonel General No Kwang Chol - one of the three promoted in a recent North Korean military shakeup.
Trump responded on Twitter, saying: 'The fact that I am having a meeting is a major loss for the US, say the haters & losers'.
The summit, which was nearly scrapped by a mercurial Trump last month, comes after a flurry of diplomatic activities and barrage of invective and insults traded between the US President and an equally aggressive Kim. Kim described the meeting as "a good prelude to peace". We'll be doing something on it.