The best approach to working your abdominals when dealing with a sensitive back is probably to avoid fully sitting up from a lying down position. There are far more effective and safer exercises to work your abs while safely supporting your back. You can still get a great abdominal workout in spite of back issues.
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Keep It Low
The best position for ab work is one where your back is fully supported against a stable surface. Lie on your back with your feet flat on the floor, hip-width apart, and walk your feet slightly in front of your knees. Keeping your entire lower back pressed into the floor, roll your head and shoulders off the floor. Your head and neck should be in a vertical position. You may place a rolled-up towel or pillow behind your lower back for support. Hold on behind your knees, extending your elbows out to the sides to draw your chest forward. Your body should resemble the letter "C." Exhale fully and hollow out your abs by drawing your navel down. Keep holding on and breathing, pulling your abs down. Repeat for eight breaths.
Get Low and Curl
Low curl is an effective exercise because it simultaneously strengthens and supports the lower back while isolating the abs in a way traditional situps cannot. The movement is precise and controlled, powered by your breath. Lie on your back with your feet flat on the floor, hip-width apart, and walk your feet slightly in front of your knees. Keeping your entire lower back pressed into the floor, roll your head and shoulders off the floor. Your head and neck should be in a vertical position. Curl forward, knitting your ribs together as if you were closing an accordion. Squeeze your buttocks to stabilize your position. Keep exhaling as you curl forward, dropping your navel. Repeat eight times, reset your position, repeat eight times more.
Use a Kickstand
Using your elbows as an anchor can help you lift your chest to better isolate your abs while supporting your back. Lie on your back with your feet flat on the floor, hip-width apart, and walk your feet slightly in front of your knees. Press both elbows into the floor, keeping them close to your sides and positioned behind your shoulders to draw your chest forward. Keeping your lower back and ribs pressed into the floor, roll your shoulders and upper ribs off the floor. You may place a narrow pillow at mid-back range for support. Curl forward in this position, trying to close the gap between your upper and lower ribs. Repeat eight times. Reset by drawing your elbows back again, lifting your chest and squeezing your buttocks. Do two more sets of eight reps.
One of the best ways to balance strength between the front and back of your body and effectively work your abs is the front plank. Start with your forearms parallel or elbows turned out, hands clasped. Extend your legs long and keep them about hip-distance apart, resting on your toes. Keep a slight posterior tilt or tuck of your tailbone to take the sway out of your lower back. This will protect your back while holding the position. Focus your gaze down to keep your neck in line with your spine. Finally, keep your abdominals pulled in. Your body should form one straight line. Widen your shoulders across your upper back, hollowing out your shoulders as you extend through your head. Hold for 30 seconds, and work up to a full minute.