One of the most important "core" muscles is the deepest layer of abdominal muscles called the transversus abdominis. They are often referred to by many other names including the lower abs, the core, your inner unit, or your powerhouse. Though all these references do ring true, the transverse abdominals are very important for stabilizing the pelvis and spine and allow the body to perform at its best.
Since the transverse abdominals (the TA's) are the deepest layer of abdominals, they are closest to your skeletal system and are the concrete to your structural foundation. They wrap around the entire middle section of your body from back to front and form the abdominal wall that surrounds the internal organs. When the lumbar spine is in its natural state with a slight arched position (neutral), the TA's also co-contract with tiny muscles attached to the vertebrae stabilizing this vulnerable area. Although these muscles do not initiate movement they stabilize the pelvis and spine while doing large movements such as spine rotation, bending over, lifting objects, jumping, lunging...everything really.
When your TA's are weak or vacationing you not only leave your spine at risk but you're body is unable to optimally run. The bigger, global muscles will attempt to do the additional work of stabilizing (often failing at doing so for sustained periods) and are put under excess stress.
Get you TA’s involved…first try lying on your back, get in neutral pelvis (tailbone should be on the ground and pelvis level), and find your hip bones. Take two fingers and move them 1 inch towards each other and down a little. Try to tighten your lower abs against your fingers while you exhale, like you're blowing up a balloon. There shouldn't be any structural change to your body, don’t use your legs or glutes, pelvis stays still. Your fingers should find a wall of resistance when you're engaging your TA's. If you're still having difficulty try coughing or laughing (you might already be doing the latter)!
Once you've figured out where those darn things are and somewhat mastered engaging them, trying incorporating breathing and engaging your TA's while doing a daily activity (I always suggest brushing your teeth since this can be increased to the twice daily routine). Here's a video to help:
In Pilates, you should not only learn how to engage your TA's but but also how to initiate, sustain and keep them active. The best Pilates keeps your TA constantly engaged, continually challenged, and motivates you to apply what you learn in the studio to your everyday living...achieving better golf swings, physically painless days picking up your kids, tougher workouts, and an even better posture!