US President Donald Trump and North Korean leader Kim Jong Un dove into the details of nuclear negotiations on Thursday against a backdrop of swirling questions about what Kim was willing to give up and what Trump may demand in the face of rising domestic turmoil. Tempering expectations, Trump opened by declaring, "There's no rush. We just want to do the right deal."The two men continued to offer hopeful words as talks began anew at their second summit about curbing Pyongyang's pursuit of nuclear weapons, a problem that has bedevilled generations of leaders. In a sharp break from his rhetoric a year ago, when he painted the threat from Pyongyang as so grave that "fire and fury" may need to be rained down on North Korea, Trump made clear he was willing to accept a slower timetable for denuclearisation."Speed is not important," Trump said. "What's important is that we do the right deal."Accompanied only by translators, the unlikely pair - a 72-year-old billionaire and a 35-year-old reclusive autocrat - displayed a familiarity with one another as they began the day's negotiations."The relationship is just very strong and when you have a good relationship a lot of good things happen," said Trump. He added that, at their opulent dinner the night before, "a lot of great ideas were being thrown about." He offered no specifics.Kim, for his part, said "I believe by intuition that good results will be produced.""I believe that starting from yesterday, the whole world is looking at this spot right now," he said, via his translator. "I'm sure that all of them will be watching the moment that we are sitting together side by side as if they are watching a fantasy movie."Possible outcomes could include a peace declaration for the Korean War that the North could use to eventually push for the reduction of US troops in South Korea, or sanctions relief that could allow Pyongyang to pursue lucrative economic projects with the South. Sceptics say such agreements would leave in place a significant portion of North Korea's nuclear-tipped missiles while robbing the United States of its negotiating leverage going forward. Asked if this summit would yield a political declaration to end the Korean War, Trump told reporters on Wednesday: "We'll see."Turmoil in WashingtonThe president's schedule on Thursday promised a "joint agreement signing ceremony" after the meeting. But as has happened before for Trump, the effort to achieve a grand foreign policy achievement unfolded against a backdrop of tumult and investigations at home.Hours before he sat down again with Kim, Trump's former personal attorney, Michael Cohen, delivered explosive congressional testimony claiming the president is a "conman" who lied about his business interests with Russia.The turmoil in Washington has escalated concerns that Trump, eager for an agreement, would give Kim too much and get too little in return. The leaders' first meeting in June was heavy with historic pageantry but light on any enforceable agreements for North Korea to give up its nuclear arsenal. Still, both offered optimistic words before Wednesday's dinner.Trump, unable to ignore the drama playing out thousands of miles away, tweeted that Cohen "did bad things unrelated to Trump" and "is lying in order to reduce his prison time." Cohen has been sentenced to three years in prison for lying to Congress.Some of Trump's previous overseas trips have also been marred by developments at home, including special counsel Robert Mueller's indictments last July of Russian intelligence officers who interfered on Trump's behalf in the 2016 election, charges that were filed days before the president and Russia's Vladimir Putin met in Helsinki.First summitAfter their first summit, where Trump and Kim signed a joint statement agreeing to work toward a denuclearised Korean Peninsula, the president prematurely declared victory, tweeting that "There is no longer a Nuclear Threat from North Korea." The facts did not support that claim.North Korea has spent decades, at great economic sacrifice, building its nuclear programme, and there are doubts that it will give away that programme without getting something substantial from the US.The Korean conflict ended in 1953 with an armistice, essentially a ceasefire signed by North Korea, China and the 17-nation, US-led United Nations Command. A peace declaration would amount to a political statement, ostensibly teeing up talks for a formal peace treaty that would involve other nations.North and South Korea also want US sanctions dialled 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.