The Muscles in Your Core
As mentioned previously, your core is made up of your abdominal muscles and your back muscles. Core strength reduces back pain because when your abdominal muscles are strong, your back has to do less work, and when it does have to work, it's strong enough to get the work done.
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Your abdominal and back muscles work together to keep your torso upright, so it's important to strengthen both groups. There are many ways to strengthen the core muscles, including movements that isolate one of the muscles in the group, or balance exercises, which force the muscles to work as a group. One example of a balance exercise is the plank pose. When you do a plank pose (like the up phase of a pushup), you are balancing on your toes and your hands. In order to remain straight and in a long angle from your toes to your head, your core muscles need to engage. If they don't, your belly will droop down, your back will sag, and you will rely on your arms and legs to try to hold your body up.
Other exercise styles that strengthen the core include swimming (because you are moving all of your limbs simultaneously), yoga (because you engage all of your muscles during each posture), and gymnastics (very similar to ballet in its balance requirement).
Over the past few years, more and more professional athletes, including football players and baseball players, have gotten interested in core-strength exercises because core strength is so important to overall athletic performance. When a football player has to reach up to catch a pass, he will do so with much less chance of injury if his core muscles are strong. When a baseball player has to crouch down to get a grounder, his chances of doing so with greater balance and more reach increase if he has a strong core.
Another benefit to strong core muscles is that you can move your arms and legs more elegantly and with more ease than you can with weak core muscles. Ballet dancers (once again) have incredibly strong core muscles, which is one of the reasons they can move their limbs in a way that looks so effortless. It isn't effortless, of course, but they are able to balance and lift their legs so extensively because they rely on their core muscles to remain tall.

