But those differences, especially since Mr. Trump withdrew the United States from the Paris climate accord last month, are inescapable. Ms. Merkel, already grappling with violent anti-globalism protests on streets outside the conference, has been intensely focused on divining a way to coexist with a president whose disruptive views differ so drastically from her own.
The best she has come up with so far is to cultivate a backdoor channel through the presidentâs daughter Ivanka, who tried unsuccessfully to persuade her father to remain in the Paris accord.
But Ms. Merkel is up for re-election in the fall, and challenging Mr. Trump has become essential in German politics. So Ms. Merkel, the courteous daughter of a Protestant cleric, is doing something she finds awkward: calling out Mr. Trump in public and questioning his commitment to the American leadership that Europeans had taken for granted since World War II.
âMerkel is clearly trying to figure out how to deal with Trump, and it isnât easy for her,â said Klaus BrinkbÃ¤umer, the editor in chief of Der Spiegel, the countryâs largest-circulation newsmagazine.
âShe doesnât like to make news in speeches, but publicly, sheâs been more critical of Trump than I would have expected,â Mr. BrinkbÃ¤umer said on Thursday, a few hours before Air Force One arrived in Hamburg from Mr. Trumpâs one-day stop in Warsaw.
âPrivately, the only obvious path is through the presidentâs daughter, which is why she invited Ivanka to that conference in Berlin earlier this year,â he said. âBut even that doesnât seem to be working. German diplomats still donât know who to call in the State Department on serious issues, or even who their counterparts are in the White House.â
Ms. Merkel and her tight circle of advisers had hoped that other White House officials â especially H. R. McMaster, Mr. Trumpâs national security adviser, and Gary D. Cohn, the National Economic Council chairman â would provide a more reliable conduit. But that has not proved to be the case. Ms. Merkelâs team was deeply discouraged by a Wall Street Journal opinion piece written by Mr. McMaster and Mr. Cohn in May that defended Mr. Trumpâs âAmerica Firstâ slogan, prioritizing the countryâs âvital interestsâ over international partnerships.
The relationship between Ms. Merkel and Mr. Trump has unfolded in stages, said Thomas Kleine-Brockoff, a former German government official who is a vice president at the German Marshall Fund. âAt first, I think she thought she could manage him,â he said. âAfter all, sheâs made a study of all these leaders â Putin, Bush, Obama.â
âYou could almost see her analyze Trump, run through the various scenarios and approaches for dealing with him,â he said. âNow I think she realizes there arenât really any.â
German officials were reticent when asked about possible disputes that might overshadow Ms. Merkelâs meeting with Mr. Trump. Her spokesman, Steffen Seibert, noted that there were many differences of opinion, âand it is not just with one delegation.â This is âalso why the chancellor is scheduling bilateral meetings to explore difficult themes,â Mr. Seibert said.
Daniela Schwarzer, the director of the German Council on Foreign Relations in Berlin, said it appeared that German officials were operating under the axiom of ârather than surfacing conflict, better not to say too much.â
Yet Ms. Merkel cannot afford to remain silent as Mr. Trumpâs unpopularity grows on the Continent. A former German diplomat who keeps in touch with her staff said Ms. Merkel had studied the February visit to the White House by Prime Minister Shinzo Abe of Japan, and had determined that Mr. Abeâs charm and flattery, coupled with a blunt public articulation of their differences, was the best approach.
That influenced her decision to invite Ivanka Trump to a womenâs conference in Berlin in April. Yet around the same time, Ms. Merkel began intensifying her public criticism of Mr. Trump. Her party, the Christian Democratic Union, has conspicuously dropped language from its campaign literature describing the United States as Germanyâs âmost important friend outside Europe.â
âMerkel is a contradiction because she understands that she is the most powerful figure in Europe but doesnât necessarily want to admit that,â said Jeremy Shapiro, a former State Department official who worked on European affairs under President Barack Obama. âBut I think she realizes that she needs to assert principles publicly to counter Trump.â
A few days before Mr. Trump arrived in Hamburg, Ms. Merkel took a shot at the presidentâs âAmerica Firstâ slogan, albeit in her typically muted language.
âWhile we are looking at the possibilities of cooperation to benefit everyone, globalization is seen by the American administration more as a process that is not about a win-win situation, but about winners and losers,â she told the German weekly Die Zeit.
Mr. Trump has told his staff that he âgets along fineâ with Ms. Merkel, though he finds the interactions awkward, two people close to him said.
But those grievances are not personal, aides insist. He is deeply displeased with Germanyâs policies, they say, and will continue to hammer Germany about its trade surplus with the United States and its refusal to pay what he believes to be its fair share for self-defense in NATO.
Still, Mr. Trump â who is almost as allergic to private confrontation as Ms. Merkel â entered Thursdayâs short meeting with no set of objectives apart from exiting quickly and without much controversy.
He praised Ms. Merkel for hosting the event under tense circumstances, one aide familiar with the interaction said, and participated in the bilateral meeting partly out of courtesy to her, not because he had any business to transact, the aide added, speaking on the condition of anonymity to describe a private meeting.
Still, it will be hard to avoid confrontation. Ms. Merkelâs aides have 48 hours to help produce a communiquÃ© from the summit meeting that all can accept, despite disagreements on climate change, immigration and trade.
In addition to the risk that the G-20 will end up 19-to-1 on the issue of the Paris accord, some advisers to Ms. Merkel fear that Mr. Trump will try to weaken support for the agreement, which was reached in 2015 with Americaâs backing.
âThere are various options that can be discussed,â was all that Ms. Merkel would say before the world leaders began arriving.
For all of these challenges, the G-20 gathering began on something of a high note for both Ms. Merkel and Mr. Trump.
Mr. Trump had fumbled a handshake in front of photographers during Ms. Merkelâs visit to the Oval Office in March, as the two sat uncomfortably in wingback chairs.
On Thursday, the chancellor extended her hand to the president, who answered with a firm and decisive grip of his own.