The student pilot reported that he completed a touch-and-go landing and applied full power to increase the airplane's ground speed. During the takeoff roll, the engine power increased, and the airplane drifted to the left side of the runway. The student pilot's full deflection right rudder application was ineffective, and the he reduced the engine power to idle. He applied brakes, but the airplane exited the left side of the runway, collided with a drainage culvert, and the airplane came to rest inverted. The airplane sustained substantial damage to the right wing lift strut, the vertical stabilizer, and the rudder. The student pilot suspected that a mechanical malfunction of the airplane's rudder system may have occurred prior to the accident sequence. However, Federal Aviation Administration, Aviation Safety Inspectors reported that during a post-accident airplane examination, the airplane's rudder system was operational and rudder control continuity was attained.