If the South wasn’t trying to conquer the North why was Lee in Pennsylvania? As to General Kearny I would recommend author and Kearny biographer William ‘’Bill’’ Styple. Another of my towns notables. He has written extensively on Phil Kearny.
He had to get the Federal armies out of the Shenandoah Valley during growing season, else he’d be unable to feed his own army. Moving into Pennsylvania and swinging toward Washington would pretty much ensure the Federal armies would follow. He also hoped his move would pull Federal troops out of Mississippi, as Vicksburg was vital to the Southern war effort. The Pennsylvania campaign was both offensive and defensive. No question but that he’d have taken Washington if he had the opportunity...that’s war.