In the Ashtanga yoga primary series you do yoga boat pose (Navasana) five times in a row (holding each for five breaths) and lifting up in between each one, and without resting (ideally.)
An alternative approach, particularly if want to work at intelligent hip flexor and quad control , is to practice Navasana, or boat pose, one leg at a time.
Sitting upright, reach your arms forwards and lean back while keeping your heels touching the floor. Pull one knee to your chest keeping the other heel touching the floor. Keep your ribcage still and focus on lifting the foot and then slowly pulling the lifted knee as close to your chest as possible. Pause, lower and then do the other side.
I'd suggest about 5 reps for each leg and work at gradually increasing the hold time. To give your hip flexors a break (and to stretch them out) you can do table top pose.
Next, repeat the exercise but this time after lifting the foot, straighten the knee. Release and switch sides. Here again work towards 5 reps. Work at keeping the knee close to your chest as you straighten it. Then work at gradually increasing hold time with knee straight.
Because the psoas, which attaches the femur to the lumbar spine, can both flex the hip and help twist the lumbar spine, try both adding a twist either towards the lifted leg or away from it, both with knee bent and with knee straight.
For example, first do the knee to the chest option and after lifting the knee turn your ribcage away from your lifted knee. Lower and repeat with the other leg, turning away from the lifted knee. After table top, do the knee straightening option and this time turn towards from the lifted foot.
Another option for boat pose is to use your hands to help. First straighten the knee. Pause and then grab your leg with both hands and use arm strength to pull the leg inwards while you relax your hip muscles. Then re-engage the hip muscles and release your hands. You can also do this with the knee bent option.
Using both legs at the same time you can practice yoga boat pose as follows. First lean back and reach your arms forwards. Keep your heels lightly touching the floor. Next lift your heels an inch off of the floor. Pause. Then lift the shins so that they are horizontal. Here again you can pause. Then straighten your knees. Work at gradually extending the length of time in each position, particularly the final one.
The seated get up is a way of getting into the one legged squat from a seated position. Even if you aren't interested in one leg squats this video does include tips on stabilizing the knees (at about the 5 minute mark.) Usual muscle activations for knee stability might include the quads, the hamstrings or any of the glutes. This looks at another set of muscles all together. If you like the video or find it helpful, please do share it! Thanks!
Some tips for learning how to do deep squats (without weight). The first tip is on how to stay balanced while squatting.
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Basic yoga poses: standing, sitting, arm balances, binds, twists, inversions, back bending, front bends.
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Some hip flexor strengthening exercises.
These yoga poses can be used as arm strengthening exercises.
Turn yoga poses into leg strengthening exercises using floor pressing actions, leverage and friction.
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Knee anatomy for yoga teachers looks at the bones and muscles that comprise the back of the knee in simple terms.
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Some exercises and yoga poses for working towards a lying quadriceps stretch one leg at a time.
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Learn to consciously control your quads and hip flexors with Conscious Muscle Control: Quads and Superficial Hip Flexors. This downloadable video course teaches you how to feel and activate your quadriceps (the vastus muscles) as well as the rectus femoris, tensor fascae latae and sartorius muscles.
For any calf stretch you have to bend your ankle forwards to stretch the soleus and/or gastrocnemius. How you bend the ankle forwards can make the stretch more or less effective.
Yoga for flexibility with stretches for the hips, quads, hamstrings, glutes, psoas, shoulders and arms. These yoga stretches are designed to improve flexiblity.