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EUGENE, Ore. — Julia Olson climbed the slope of Spencer Butte, taking the steeper of the two paths. Near the summit, shrouding pines suddenly gave way to a vista of the Cascades. On this day, summer wildfires, their season lengthened by climate change, put a haze in the sky.

The climb and return, which she can power through in an hour, is a head-clearing ritual for Ms. Olson, an environmental attorney. It is also a place to compose the soaring language of opening and closing arguments. “I do my legal thinking here,” she said.

If all goes as planned, Ms. Olson will deliver her opening argument on Monday in a landmark federal lawsuit against the Trump administration on behalf of 21 plaintiffs, ages 11 to 22, who are demanding that the government fight climate change. It is a case that could test whether the judicial branch has major role to play in dealing with global warming, and whether there is a constitutional right to a stable and safe climate.

The young plaintiffs claim that the government’s actions, and inaction, in the face of global warming violate their “fundamental constitutional rights to freedom from deprivation of life, liberty, and property.” Their age is central to their argument: For older Americans, the potentially catastrophic effects of climate change are a problem, but ultimately an abstract one. Today’s children, however, will be dealing with disaster within their lifetimes; the youngest of the plaintiffs, Levi Draheim, will be just 33 in 2040, the year by which a United Nations scientific panel now expects some of the biggest crises to begin.