Core Control is the optimal recruitment of the trunk musculature required to perform a task in relation to the anticipated load. In other words, Core Control refers to activating the muscles of the inner unit "as much as necessary, as little as possible."
Axial Elongation is the proper alignment of the head, spine and pelvis that provides optimal spacing of the joints during movement, and prevents compressive and shear forces that may cause spinal disease. By optimizing joint spacing the soft tissue surrounding the joint can move freely which minimizes the risk of injury.
Now that we understand what Principle 2 is, how do we accomplish it? To create axial elongation, imagine you have a bungee cord attached to the crown of the head (see posture A below). This bungee has a very slight pull on your head causing you to lengthen up and out of your feet. In fact, you may feel like your feet are barely touching the floor. Upon creating this lifting affect, the head, spine and pelvis start to find their 'neutral' position. The position where the forces on them are even from all sides. You are creating optimal space between your joints. Then we need to hold this 'neutral alignment' with our musculature. Remember Principle 1, Breathing? Core activation is a given when using the Pilates Breath as Joseph Pilates taught it: "SQUEEZE EVERY ATOM OF AIR FROM YOUR LUNGS UNTIL THEY ARE ALMOST AS FREE OF AIR AS IS A VACUUM. Stand erect (lengthen the spine) again and observe how your lungs will automatically completely refill themselves with fresh air." (page 13, Return to Life Through Contrology). Once you have created the engagement of the core musculature through your breath, you will have a stabilizing effect on the spine.
Remember, ALL movement initiates from our core. From the core, movement then extends up the spine (resulting in axial elongation or spine lengthening) and out into our activity through our extremities, at our discretion, rather than in an unaware manner. Move with intent and you will have a strong, well balanced body.
Axial Elongation places the body in its optimal position to increase available degrees of freedom and increase efficiency of movement. - Polestar Pilates™ Manual - Principles of Movement 2008