In yoga for strength an important consideration for smooth (and relatively effortless) advancement is the idea of progressions, gradually making an exercise more difficult.
In weight lifting the equivalent would be gradually adding more and more weight.
A prerequisite for both strength and flexibility is stability.
Stability generally requires using muscles against each other with the result that the opposing muscles are strengthened. And so practicing stabilizing parts of the body is actually a strengthening exercise in and of itself.
But in general stability is the practice of making a part of the body resistant to change. Then it can be used as a foundation for another part of the body to create change.
For more on stability (and strength) you can read exercises to improve stabilty.
One way to progress in the push-up action (using the arms to push the body away from the floor) is to first practice a relatively easy exercise like kneeling one armed push-ups.
Despite being a one-armed push, up this exercise is fairly easy since most of the weight of the body rests on the shins.
Another progression or preparation pose that is both a yoga strength exercise and a yoga stretch is bending the elbows in low lunge.
Initially you can use the arms to lift and lower the chest while the back knee stays on the floor. The front leg takes a lot of the weight of the body off of the arms.
To add more weight lift the back knee. You can either lift the back knee before bending the elbows or after. Or you can keep the back knee lifted throughout this yoga strengthening exercise. In any case lifting the back knee can be used to increase the amount of work that the arms have to do.
But all of this is for naught if you haven't got the flexibility in the hips. And so part of this exercise can be learning to relax the arms and allowing the ribcage to sink down. This stretches the hip extensors, particularly if you can lower the ribcage below the height of the front knee. You can then strengthen the arms when pushing the ribcage back up.
As for the actual yoga push up, one way to progress in this yoga strength pose is to first practice lifting just the ribs.
Initially lift the chest off of the floor. Work at gradually pushing the chest higher.
Then you can work at lifting the hips independently, and then the knees. The yoga push ups article explains it in more detail.
One other possible yoga for strength exercise that also works on flexibility is the half-split one-armed-crocodile (Chaturanga) exercise.
While in half split with the chest and hands on the floor, lift the chest using both arms. Prior to lifting one arm, add tension to the straight leg. Then add tension to your abs. Then make the bent leg arm strong. Relax your straight leg arm. Then lift it and reach it to the side. To rest, put the hand down on the floor, and then lower the chest to the floor.
You can also do the same when working towards full side splits.
An option (or progression) is to start with both elbows on the floor. Make the bent knee arm strong and lift the other arm. Then place the arm back on the floor to rest. Then repeat.
This not only strengthens the arms (and helps to balance the strength in both) it also increases the stretch of the inner thigh. And it strengthens the inner thighs at the same time.
Another pose for strengthening the arm statically is side plank yoga pose. If your balance isn't so good then first practice balancing in side plank. You can first practice shoulder stabilization while resting on your elbow. In side plank you can try different variations to help strengthen your outer thigh, your inner thigh, your quads and even your hamstrings.
One way to improve strength while doing yoga poses is to practice moving in and out of yoga poses, particular those where the arms or legs bear the weight of your body.
One of the harder exercises for the arms is lifting up into wheel pose.
You can practice holding this pose to develop strength or practice lifting and lowering repeatedly.
Details on learning how to lift the head are covered in the pdf Making Wheel Pose Easier.
For strengthening the legs, one of the more difficult exercises is the one legged squat or "pistol" squat. To work up to this, first work at doing a regular squat on both legs.
From standing, see if you can lower to half way (bottom left), then stand up again. From a full deep squat (below right) you can also lift into the half squat.
In both the deep squat and the half squat you can then work at shifting weight on to one leg. Work towards this gradually. Then once you can get all of your weight onto one leg see if you can lift the other leg.
In a deep squat, if you can get all of your weight onto one leg, then reach the other leg forwards.
You can first practice standing up from this position, then lower using both legs. Once you are comfortable standing up with a once legged squat, then work at lowering down on one leg.
A pose that can be used to work on the back of the legs and or the arms is reverse plank.
To strengthen the hamstrings try these hamstring strengthening yoga poses.
The back can be divided into an upper and lower half. Generally the upper back is the back of the ribcage while the lower back is the part that connects the pelvis to the ribcage (the lumbar spine.)
Upper back exercises include exercises for becoming more aware of the thoracic spine and ribcage and also training the spinal erectors to bend the ribcage backwards.
Low Back exercises (for dealing with low back pain and helping to prevent it) include a mix of low back strengthening and hip strengthening exercises.
One of the simplest exercises for the abs is to slowly sit up and balance on the butt from a laying down position. With arms and legs reaching up, slowly reach arms and legs forwards. Then when you feel your center over your buttocks, slowly sit up.
To make it easier grab onto your thighs.
Then lay back down slowly and in control and repeat.
A progression is to roll up to a squat. (First do it fast to get into the squat, then see if you can slowly roll up into a squat.)
Then you could also try rolling up into a one legged squat as above.
This yoga routine video is designed to help you strengthen your arms and legs via the use of friction and pressure. It also teaches you how to become more aware of your body.
Kneeling, lying, standing and wall assisted quadricep stretching yoga poses with tips for making these stretches more effective.
The transverse abdominis muscle can be broken down into three parts. Transverse abdominal exercises can thus affect the SI Joint, lumbar spine and the lower portion of the ribcage.
To improve the resiliency of your knees it can help to exercise them in a variety of positions. The following yoga poses can be used as knee strengthening exercises. The trick is to activate your knees while doing them.
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.
Glute and Hamstring activation can be used to compliment the quad and hip flexors for a balanced practice. Conscious Muscle Control: Hamstrings and Glutes is a video course designed to teach you how to activate your glutes and hamstrings at will. You'll also develop the ability to feel them activate and relax.
Learn how to activate your quads and hip flexors so that you can use them at will. Conscious Muscle Control: Quads and Superficial Hip Flexors not only teaches you how to activate and relax your quads and hip flexors at will, it also teaches you how to feel when they are active and when they are relaxed. This clearly defined awareness can help you get more in touch with your body.
Make your yoga poses less wobbly with less effort. Grounding and centering are two techniques for creating stability in yoga poses.
Arm supported yoga poses can be used to strengthen the arms and shoulders. Includes plank, chaturanga dandasana, downward dog, dolphin pose, side plank, wheel, reverse plank, table top pose.
Exercises in muscle control 1 teachers you how to activate and relax your knees, hips, front and back of the leg and also inner and outer thighs. These activations can be used in standing poses as leg strenghtening exercises and to improve flexiblity.
The transverse abdominus muscle can affect the SI joint, lumbar and lower thoracic spine stability, used in various diaphragmatic breathing techniques and act as a tension adjuster for the rectus abdominus.
Effectively Activating Transverse Abdominus can mean better stability for the SI Joint as well as for the lumbar and lower thoracic spine.
Rather than fighting through joint pain here is an overview of the approach that I've used to help alleviate hip pain, knee pain or shoulder joint pain while doing yoga poses.
Make balancing easier. Use pressure sensitivity to feel your center of gravity.
Camel Yoga Pose or ustrasana is a kneeling pose that can be used to stretch the hip flexors. One key action that may help in getting your pelvis forwards more is pushing your hands forwards, either against your feet or against the floor.
A yoga approach to how to do squats including how to stay balanced, and avoiding knee or hip pain even while going all the way down.
The transverse abdominis can have an affect on sacroiliac joint stability as well as stability of the lumbar spine and the T12/L1 junction.