Ample time for recovery -- especially after a big exertion such as that required for a race -- is an essential part of a balanced running schedule. The days that immediately follow a race can include some activity, but the main idea is to allow the body to recoup. Distances up to and including a half marathon require less rest than a marathon, but taking some down time can get you back to a full training regimen faster than trying to push your body too much in the days after a race of more than 13 miles.
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If you have traveled a long distance in order to race, you may not have the time to run the next day, or you may be too tired from the travel. Try to take short walk or stretch breaks so that your muscles don't get stiff and sore. If you are traveling by public transportation, stand up and stretch or walk down the aisle every 20 minutes or so.
Runners who are used to high-mileage weeks may run the day after a half marathon. Do so only if you are not excessively sore and can keep the pace at a rate that doesn't interfere with your recovery. Jogging or walking mixed with jogging for 15 minutes is plenty if you feel the need to run. For most people, however, resting or active rest is a better option. If you experience excessive soreness or pain during your recovery run, stop and walk back home. There's no need to push though a recovery run and risk injury.
Pool running is one of the best cross-training exercises for runners, as it's non-weight bearing. The cool water feels good on muscles -- especially if they are sore -- and you can simulate the motion of running. This movement will increase blood flow to the muscles and aid in recovery, but it does so without the added stress and impact of running on dry ground. Run in the pool or swim several days until your muscles are no longer sore, and then ease back into running on solid ground.
In addition to resting after your half marathon, you should focus on good nutrition to facilitate your recovery. Aim to consume 3 to 5 grams of carbohydrates for every pound of body weight. Eat at least three well-balanced meals with plenty of protein the day after your race and drink plenty of fluids to stay hydrated. If you can, splurge on a session of massage therapy while you enjoy thinking about your accomplishment of completing a half marathon.