Yoga is a form of mediation which involves physical exertions as well as inward examination. Is this an acceptable form of meditation for a Buddhist? Are there any sects of Buddhism which focus on meditations which involve physical exertion?
It depends on what your definition of 'acceptable' is. Will it benefit you? Absolutely. For one, a case can be made that yoga is a mindfulness-of-the-body meditation and that it can even go beyond that - incorporating other aspects of sati (mindfulness). I think it's also safe to say that all five of the hindrances can be temporarily suppressed during intense yoga and may even give rise to access concentration (upacarasamadhi).
Still, I don't think it's a perfect substitute for the kind of meditation the Buddha taught. For starters, I wonder whether true jhana is available to someone in an asana other than a variant of the lotus position. You'll be able to attain some amount of samatha, but the degree will be lessened by the effort it takes to maintain the other asanas. Outside of lotus, I also don't think you'd be very stable, physically speaking, and that you'd probably just wind up falling over! But maybe you aren't even looking to attain jhana. Who knows! There is precedent that such states aren't a prerequisite to enlightenment.
As per 'sects' of Buddhism that use physical excursion, well, there is a rich tradition of doing prostrations, schools like Tendai have their mudras (but I'm guessing that's not the kind of exertion you're talking about), and many martial arts schools with a Zen slant pay strict attention to bodily position - especially Japanese styles like iaido (and come on - meditate with a sword? How cool is that!).
So yes - beneficial, absolutely. A true alternative? Mmmm...probably not.
Asanas are generally acceptable if you leave the ideological side of it and when you do the Asana be aware of the posture and sensation in the body. For timing purposes use the breath and heartbeat. This can be a good physical exercise also converting this to Kayanupassana and Vedananupassana in the 4 Satipatthana.
If you leave out the ideological part behind Pranayama then both are compatible. The distinction between the two are:
- Buddhism: do not regulate the breath but watch the process initially but later slow the process
- Pranayama: regulate the process
Some buddhist tradition sometimes do advice to regulate the breath to overcome being scatterbrained when meditating, being lethargic, and to induce pleasantness / joy. Also there is a view that Pranayama could have been originating or evolution from a Buddhist teaching. (See: Can Pranayama Be Derived from Anapana?)
Also when doing Breath Meditation some traditions mention certain prominent places to rest your attention. If you go by the Suttas only it say to rest the attention in front or around the mouth as some traditions interpret. (See Parimukhaṁ). This can be analogous to Chakras in Yoga. (See: Are the Chakras present in the original Buddha's teachings?).
Whatever you do start with at then end you have to see things as they are. For this you have to reduce the of the breath and develop strong awareness. If you are in regulating the process you lose the chance of being objectively observing and being aware. The awareness you develop would be shallow. This might be a hindrance later on. But if this is not the only techniques you use then it can become a tool. In addition you are creating new fabrication / conditioned existence where the objective was to get out of it. Having said that the path leading to the final goal is actually fabricated / conditioned. Sometimes good conditioning can help.
Having said all this do not give too much prominence of time to Yoga though it can be used for benefit as time is limited but use this as a tool in certain situation like to overcome lethargy, develop energy, overcome pain, etc.
For an alternate discussion see: Yoga: as seen in the light of Vipassana and A Re-appraisal of Patanjali’s Yoga-Sutras in the Light of the Buddha’s Teaching
Are you asking whether or not Buddhists can practice Hatha Yoga (i.e. practicing the physical postures) or the yogic meditative practices? I don't see any real problem with Hatha Yoga, but yogic meditation is based on a different set of teachings and is part of a different interpretive framework than Buddhism.
Buddhism rejects the idea of any self that exists other than as a convention and teaches that all of reality arises from causes and conditions without any underlying ground of reality.. Raja Yoga (the proper name for Yoga as a meditative system) is based very much on the exact opposites of these, teaching the existence of a supreme cosmic principle underlying all things called Brahman, and furthermore most teach that this divine Brahman is the true self, or Atman.
The actual meditation methods used in Raja Yoga are effective means of attaining concentration, but the purpose of attaining such concentration is quite different in Buddhism. In Buddhism, concentration is used to clear and prepare the mind for the realization that reality is impermanent, unsatisfying, and non-self, which leads to the attainment of enlightenment, and so concentration is combined with other methods which aim at this goal. In contrast, Raja yoga identifies the attainment of high levels of concentration (i.e. the Jhanas) with the realization of the Atman as being Brahman.
From a Buddhist perspective, such an interpretation is wrong view, and especially coupled with the fact that Raja Yoga doesn't have practices leading to the realization of the three characteristics, Raja Yoga isn't a system of practice which one should expect to lead one to Nibbana.