What did lee hope to accomplish by invading the north?

Confederate general Robert E. Lee hoped for the destruction of the Union Army and subsequently Lincoln's surrender.

Lee knew he was a superior field commander and had greatly superior team of generals working for him at Gettysburg. He also knew that the greatest part of the Union army was in the North so he wanted to destroy it and then head towards the capital for Lincoln's surrender.

by 1890, the united states had by far the world's most productive economy. american industry produced twice as much as its closest competitor--britain. but the united states was not a great military or diplomatic power. its army numbered less than 30,000 troops, and its navy had only about 10,000 seamen. britain's army was five times the size of its american counterpart, and its navy was ten times bigger. the united states' military was small because the country was situated between two large oceans and was surrounded by weak or friendly nations. it faced no serious military threats and had little interest in asserting military power overseas.