Sorry this post is a bit all over the place. I always have the urge to talk about 10 things at once and I don't feel like splitting this post into three, so here they are, a whole bunch of topics in one. Feel free to read and comment on only parts of this post.
Kino's video on "Accessing Forward Bends in the Ashtanga Yoga Primary Series" is a great reference video of the foundations of Ashtanga - the tristana: breath, postures, drishti. She offers excellent explanations of Ujjayi, or rather, "free deep breathing" (3:04 in the video), as well as the mula bandha, and the mechanics of a forward fold. I learned a lot from this video. Some yogis (including me) like to close their eyes during practice. Grimmly also questioned when dristi came about in the traditions. I can't comment on that because I am not familiar with the sutras or the history of yoga (I am not a diligent enough yogi >_<~), but at least Kino cleared up the purpose of the drishi, i.e., the energy goes where our gaze go. So if we look down in our forward folds, we probably bend downwards instead of forwards.
Now for a little bit of controversy. David Robson blogged about how Sharath mentioned at a conference that ujjayi is a pranyama technique to be practiced once a student reaches advanced series and not part of the Primary series asana practice. Say what? I guess there's no name to the breathing we do in Asthanga. The new official name is "free deep breathing (with sound)", like Ujjayi but not actually Ujjayi. This was in the comments section of the blog:
"A student had once asked Guruji if the breath we do during practice is like Ujjayi. Guruji didn't give a clear answer (I picture the Indian head bobble), and so the misnomer was born."
That wasn't too bad unless you're not used to being anal about the correctness of labels.
Now, mula bandha: Michael Stone (a psychotherapist who studied Ashtanga with Richard Freeman, but also studies other styles of yoga, plus he can read Sanskrit) gives his interpretation of mula bandha at about 5:25 in this video: The inner traditions of yoga. I highly recommend watching the whole video by the way. Unlike Kino who went into detail on the anatomy of mula bandha, Michael talked about not squeezing the anus or the pelvic floor, but rather a natural toning of the pelvic floor at the end of an exhale. The "bandha" is not really a lock but a yoking of attention and the feeling of the breath (he describes prana not as the breath itself but a feeling pattern of the breath, a life energy that's associated with the breath) at the end of an exhale. To be honest, I find it hard to focus at the end of the breath (still a beginner), but I understand why he teaches it this way. Mula bandha is probably a pretty subtle engagement, but as a beginner practitioner, it's hard to engage the right amount of muscle fibers, so students were just taught to "squeeze the anus". If you squeeze everything in that general area the mula bandha will be engaged. However like Michael says, it's pretty uncomfortable and students forget to breath. That I agree with from experience. I think with Michael it's free breathing first, the bandha comes later.
In my practice today I experimented with both "general area squeeze" and no squeeze in various asanasa + vinyasas, and found that Michael was correct in that it's not necessary to do the massive anal squeeze to engage the core. Poses that require a lot of core (e.g. utpluthih), muscles in the core area get engaged anyways, otherwise the pose won't happen. Mega-squeezing did actually induce extra unnecessary strains. On the other hand I noticed it was really hard to keep the uddiyana bandha engaged at all times. I think with more practice I'll be able to engage the right set of muscles; for now I'll go easy with the "mula bandha overs-queezing", and focus on the uddiyana bandha instead.
Nobel wrote two posts on the topic of core-centered practice. My take is that for Ashtanga yoga and utpluthih (probably some other poses too) specifically train for core strength. Jump throughs are not possible without hardcore core engagement. It is totally possible to do a lot of the more basic poses without bandha engagement, e.g. warrior II, tree pose etc. However, no core engagement makes it really easy to get injured. Coming up from a standing forward fold is unsafe when core is not engaged to protect the spine. So is upward facing dog. In fact, my colleagues recently strained their lower backs from doing YogaX (a power yoga sequence in P90X) because Tony Horton didn't emphasize the concept of "sucking in the belly button", a.k.a. core engagement in upward facing dog or coming up from standing forward folds. People love him, and macho people who would otherwise never try yoga in a million years are practicing yoga because of him, so I guess I shouldn't judge. Nobel asked if bandha (core) control is necessary for safe yoga practice, how should beginners approach yoga then, if they start out having little core strength? That's when I think pilates classes come in handy. I do think it's a good idea for people to build some basic core strength through pilates before getting into power / Ashtanga yoga. However, I noticed some people in our studio would only take pilates classes and avoid yoga classes. I had a discussion with a yoga teacher friend who thinks it's because pilates classes offer straight-forward feedback - "feel the abs burning!" whereas yoga requires more body-limb coordination and upper body strength. People who lack upper body strength and who only want the satisfaction of "getting a good work out" minus the frustrations of a learning curve may prefer pilates over yoga.
Anyways, it sounds like I'm becoming totally judgmental, but I just find it fascinating trying to figure out why some people will only do restorative and yin yoga, some will do 2 hot yoga classes in a row every day, some will only do pilates, and others will only do power yoga. Hmm.. and which of these people have the strongest vs. the nicest looking core muscles? :D