When dealing with small infestations, mechanical control techniques like hand-pulling, cutting, or brush hogging can be effective. These techniques require more work, especially for long-term control, but they can reduce or eliminate the need for herbicide.
This 4-page article from the New England Wild Flower Society reviews common invasive plant control methods, with mechanical methods discussed on the first two pages.
It’s hard to remove invasive trees and shrubs by hand. Tools like the Pullerbear help get these plants and their roots out of the ground.
This longer video describes how to use a bush hog, a tractor attachment for cutting larger brush up to 2 inches in diameter. Jump to 12:43 to see the attachment in action.