I, like most runners, tend to focus more on training and forget that some strength exercises are required to reach your performance peak.
What are some good exercises that you can do at home without gym equipment?
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To me core training is very central. Your core muscles includes all the main muscle groups in you torso and increasing the strength of these will not only help with your running but also with all sorts of back problems if you have a desk job like me. See about.com for a pretty good introduction to core training. Very few core exercises requires any equipment.
In my experience, strength training for distance running has two major benefits:
maintain the proper form throughout the run, when not running in proper form, the running efficiency decreases dramatically;
help running uphill and speed up in the last sprint to the finishing line;
For 1, the core training is essential, and you can find quite a few training tutorials at runnersworld.com, they have great videos demonstrating core training from beginning to advanced level, and those all can be done without gym facilities;
For 2, workouts that build your overall body strength are enough, like some variations of deadlift as they work on from thighs and calves to back and to shoulder and arms, and sometimes two dumb-bells will do.
I've ran twenty marathons and ultras, so far without injuries. I think this is because I do a lot of strength training.
The simplest is walking upstairs and downstairs. My building has 30 floors, and walking up and down 3-4 times is a nice workout in bad weather. When I walk upstairs, I step on the balls of my feet only, which better develops my calves and ankles.
Also I do a lot of core training, including kayaking - it develops pretty much the same muscles as sit ups, and is more scenic.
The core is ideally vertical and inclined to motion direction. This requires stabiliser muscles in core and legs. Often, you see people in marathons who need to walk backwards with intensive pain in knees. This is because they have overused their forwards muscles or undertrained some parts of the body such as the other side of the torso like back. The front thighs, shins and calves are undertrained while hamstrings are overtrained only in one direction.
The body is meant to move in all kinds of directions, twisted and rotated.
The other side of the legs can be worked with trainings such as backward movements, some instructions here where backwards motions work on the stabilisers on the other side of knee and core (this can be useful particularly if pain developed in knees over last part of the run).
Ankle stabilisers can be worked out with plie
Femur stabilization can be worked out with recreational activities such as different boarding activities (skateboarding) and dancing
Vertical motions such as stairs
Depending on your body alignments such as feet pronation, twisting the torso and adding more tempo with arms and legs may be done but they require particularly core strength
Working on the deep psoas muscle group, alternating deep breathing with short breathing such as 3short-2deep rhythm while running (less injuries in the left knee).
Optimal running is inherently a periodic twisting motion where left side is stabilised with right side: the body acts like a spring where the elastic energy is reutilised from the muscles. Any stiffness or shortcoming can make the running more inefficient. Cross-training other things with twisting such as kayaking, speed skating, kickboxing, dancing, ballet and soft-sumo squatting of proper Powerlifting technique can add more strength and joy to the training.
Do not proceed to too advanced technique such as gazello running-style too quickly. Faster speeds require more from the core and leg. An undertrained body can compromise your cardiovascular health. Advanced technique is built slowly where the strength is built first and slowly your cardiovascular potential is developed to higher level. Developing deep muscle layers is vital to optimal breathing technique and less injuries.
Enjoy your training.