The student pilot reported that he was performing a touch and go landing during a solo traffic pattern flight and had added full engine power for takeoff when he noticed the airplane had drifted right of the runway centerline and made a correction back by applying left rudder. The airplane then began oscillating and increased with each control input. Realizing that the airplane was not going to be able to takeoff safely, the student pilot aborted the takeoff. He continued to apply corrective directional control inputs, but the airplane departed the runway surface and impacted terrain in a field adjacent to the runway. The airplane sustained substantial damage to the right wing. The student pilot reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation.