What is a good core exercise besides sit-ups?

Try slow mountain climbers (for those with no shoulder or wrist issues). Start in the push up or plank position, straight back, firm core, holding in the abdominals. Breathe out while bringing one knee inward, then straighten leg, breath in. Alternate knees in a climbing motion. Try doing this for one minute.

This is a challenging move that will work the entire core!

There are so many good ways to train the core.  I am not always a big fan of sit ups because it can cause extra stress to your back.  I like to perform roll downs--kind of a backwards sit up. Start by sitting on the floor (mat) with your feet together, knees bent and together, head pulled up and shoulders back.  Keeping your belly button pulled toward your spine, roll down until your feet feel like they are coming off the mat.  Hold for 2 seconds and come back up to the sitting position.  The stronger your core becomes the further you will be able to roll down while keeping your feet flat on the mat.

Also don't forget your core is like a weight belt surounding you, so work it in all directions.  If you do a plank (resting on your elbows and toes face down) also do a plank in a side lying position (being sure to work each side) and face up (on elbows and heels) as well.
When you start training your core you want to work the stabilization musculature. These exercises help stabilize the spine and pelvis.  Here are some examples of core stabilization exercises: 
Lie on your back with your knees bent, feet flat and toes pointing straight ahead.   
Draw in your belly button and squeeze your glutes. 
Lift your pelvis off the ground until shoulders, hips and knees are in line. 
Slowly lower pelvis to the ground. 
Exercise should be done slowly maintaining belly button draw-in at all times.
Prone Iso-Ab 
Lie on your stomach with forearms on the ground. 
Draw-in your belly button 
Squeeze glutes to raise yourself off the ground 
Hold yourself there for as long as you can maintain proper form up to 60"
Once these exercises become easy, you can lift one leg or try doing them on a stability ball. 
BOSU Planks with variations are great core exercises. These challenging core exercises are great for runners. Starting Position: Flip the BOSU over so that the hard, flat surface is facing up. Place hands on top of the BOSU, at least shoulder width apart or hold onto the side handles on the BOSU. Make sure your shoulders are directly above your wrist and that the rest of your body is in a straight line from your head to your heels. Pull your belly button in towards your spine to prevent the lower back from doing too much work. It is helpful to have a mirror to watch your form. Movement: Press up into plank and hold for as long as you can. Variations: When a one minute hold has been mastered, lift the right leg up while holding plank, lower with control and switch to the left. You can add movement with the upper body by rocking the BOSU board back and forth and/or side to side. Once you do add movement, be careful not to lose the stability in your core (belly button in) or in your alignment. Repetitions/Time: Start with holding plank from 10 seconds to a full minute or longer if you choose. Once you can hold plank for one minute start adding movement such as lifting one leg at a time or rocking the BOSU with the upper body.

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