"If Kim Jong Un jumped up on the table, pulled down his trousers and took a dump in front of Trump", said Lewis, "the president would tell us that he was especially impressed with a private but moving gesture made by Kim".
On the agenda, trying to iron out the next steps in the denuclearization of the Korean Peninsula.
Trump, who landed in Hanoi at about 9pm local time (14:00 GMT), is seeking an assurance from North Korea that it will stop testing its nuclear weapons and ballistic missiles.
In contrast to Trump's direct route to the summit, Kim embarked on a two-and-a-half day, 4,000 kilometer (2,500 mile) odyssey through China by train.
"This is where the president's unpredictability, his impulsiveness, his inclination not to prepare for meetings could get us into trouble", said Victor Cha, the Korea Chair at the Centre for Strategic and International Studies, whom the Trump administration considered nominating for ambassador to South Korea.
After an initial historic meeting in Singapore in June that produced only a vague statement about denuclearization, analysts say the second date must deliver more in the way of tangible progress.
"A lot of things are going to be solved I hope", Trump said as dinner began. And while North Korea has now gone more than a year without conducting missile and nuclear tests, it has done nothing to roll back the weapons already built. "His words are followed by actions", said one admiring Hanoi resident, Le Dinh Hung, who carried a painting of Trump that he wanted to give him.
North and South Korea also want USA sanctions dialed back so they can resurrect two major symbols of rapprochement that provided $150 million a year to the impoverished North by some estimates: a jointly run factory park in the North Korean border city of Kaesong and South Korean tours to the North's scenic Diamond Mountain resort.
Morgan Stanley said it estimated the opening up of North Korea's economy, assuming a gradual Vietnam-style liberalization, could bring investment opportunities of up to $9 billion per year and incremental consumption opportunities of $2 billion per year.
Kim prefers to meet with Trump rather than his staff, with North Korea accusing Mr Pompeo of having a "gangster like" attitude at a meeting following the first summit, before calling a later meeting in NY.
That different negotiating approach has raised concern that Trump, anxious for a deal, will give away too much to North Korea, which has a history of backing out of agreements. For that reason, he said, a formal peace declaration ending the Korean War ― which would supplant the armistice signed in 1953 ― would be ill-advised.
While he is in Hanoi, his former personal lawyer Michael Cohen is testifying before USA congressional committees, with the president's business practices the main focus.
Mr Kim could agree to allow a U.S. diplomatic liaison office in Pyongyang, a USA goal dating back to Bill Clinton's administration.