HAMBURG, Germany — Over her 11 years in power, Chancellor Angela Merkel of Germany has proved uncommonly adept at solving the puzzle-box challenges posed by the world’s most unpredictable leaders. But she has never met a problem like Donald J. Trump.

Mr. Trump and Ms. Merkel — estranged by widely diverging temperaments, worldviews, leadership styles and visions of Europe — had a breakthrough of sorts just before their brief meeting on the eve of the Group of 20 conference in Hamburg on Thursday night. They got the handshake right.

The 40-minute meeting that followed was mostly uneventful, touching only lightly on the hot-button topics of climate change, trade and “burning foreign policy questions” on North Korea, Ukraine and the Middle East, according to a brief statement from Ms. Merkel’s government.

People close to these two most powerful Western leaders say the brevity and bonhomie were, in fact, the main goal. Both sides are hoping for a series of low-octane interactions in which the two articulate their differences without the awkward optics of previous meetings.

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 president’s 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.

“Merkel is clearly trying to figure out how to deal with Trump, and it isn’t easy for her,” said Klaus Brinkbäumer, the editor in chief of Der Spiegel, the country’s largest-circulation newsmagazine.

“She doesn’t like to make news in speeches, but publicly, she’s been more critical of Trump than I would have expected,” Mr. Brinkbäumer said on Thursday, a few hours before Air Force One arrived in Hamburg from Mr. Trump’s one-day stop in Warsaw.

“Privately, the only obvious path is through the president’s daughter, which is why she invited Ivanka to that conference in Berlin earlier this year,” he said. “But even that doesn’t seem to be working. German diplomats still don’t know who to call in the State Department on serious issues, or even who their counterparts are in the White House.”

Ms. Merkel and her tight circle of advisers had hoped that other White House officials — especially H. R. McMaster, Mr. Trump’s national security adviser, and Gary D. Cohn, the National Economic Council chairman — would provide a more reliable conduit. But that has not proved to be the case. Ms. Merkel’s team was deeply discouraged by a Wall Street Journal opinion piece written by Mr. McMaster and Mr. Cohn in May that defended Mr. Trump’s “America First” slogan, prioritizing the country’s “vital interests” 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. “At first, I think she thought she could manage him,” he said. “After all, she’s made a study of all these leaders — Putin, Bush, Obama.”

“You could almost see her analyze Trump, run through the various scenarios and approaches for dealing with him,” he said. “Now I think she realizes there aren’t really any.”

German officials were reticent when asked about possible disputes that might overshadow Ms. Merkel’s meeting with Mr. Trump. Her spokesman, Steffen Seibert, noted that there were many differences of opinion, “and it is not just with one delegation.” This is “also why the chancellor is scheduling bilateral meetings to explore difficult themes,” 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 “rather than surfacing conflict, better not to say too much.”

Yet Ms. Merkel cannot afford to remain silent as Mr. Trump’s 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. Abe’s 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 women’s 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 Germany’s “most important friend outside Europe.”

“Merkel is a contradiction because she understands that she is the most powerful figure in Europe but doesn’t necessarily want to admit that,” said Jeremy Shapiro, a former State Department official who worked on European affairs under President Barack Obama. “But I think she realizes that she needs to assert principles publicly to counter Trump.”

A few days before Mr. Trump arrived in Hamburg, Ms. Merkel took a shot at the president’s “America First” slogan, albeit in her typically muted language.

“While 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,” she told the German weekly Die Zeit.

Mr. Trump has told his staff that he “gets along fine” 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 Germany’s 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 — who is almost as allergic to private confrontation as Ms. Merkel — entered Thursday’s 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. Merkel’s aides have 48 hours to help produce a communiqué 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 America’s backing.

“There are various options that can be discussed,” 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. Merkel’s 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.