Why, on my 2001 pt cruiser, does transmission shudder and stick in first gear

Hello, That is a sign that the clutch packs inside the transmission have worn out. There may be other factors associated with the issue, but typically when the transmission shudders and fails to accelerate your transmission is slipping. The reason it may not be shifting out of first gear is likely associated with the electronic control system for the transmission (part of the PCM or a TCM). If the transmission control module us seeing incorrect output shaft speed for the input shaft speed, it may choose not to shift. You should also have a check engine light on, or some other warning telling you to inspect the vehicle/drive-train. All that being said, I still would not rule out a possible issue with the engine (fuel delivery or ignition components), and I highly recommend a professional inspect the vehicle and diagnose the issue.