I'm curious what you are taught about martial application for slow movements?
Is there a general concept that a slow movement has a particular meaning? Or does it really depend on the specific movement within the form?
What about the execution of the slow movement: is it done relaxed, or is there dynamic tension? For me, depending on the school I'm practicing at, it seems it can vary.
Here are three examples of forms that have slow movements within:
The answer to this question entirely depends on the form and the history behind it.
Most forms have changed in small and often big ways since they were first created. Sometimes movements are repeated slowly just because someone thought it would look better in a demonstration. That's the truth for most of what you'll see out there when you see something being done slowly.
Dynamic tension is an exception. There's an actual purpose behind it. It's done slowly not for the sake of looking slow, but because you're tensing up your muscles and feeling the burn, basically. You have to go slow for that. How slow? It's timed with breathing, so that you perform the movement in one breath. And your breathing is also slowed, due to abdominal muscle contraction. I won't explain dynamic tension, but suffice it to say, there's a reason for it and why it's done slowly.
In the TKD video you linked to in your question, it shows a form where a kick is done slowly. This is done in TKD as a test of balance, flexibility, and your ability to remain calm. You generally don't see this in older martial arts, because that wasn't something they cared about as far as forms are concerned. They might have developed those attributes in other ways, just not in forms, because forms had another purpose.
Other times in forms you might see slowness done with little or no tension but with a coordination of breathing. That could be added after the fact (after a form was created). And it's usually added by someone who wants to incorporate a form of chi-kung / ki-ko to their martial arts. They're attempting to circulate chi/ki to their limbs. You'll see that a lot in Chinese kung-fu forms.
Slowness that's not due to dynamic tension, chi-kung, or as a test of balance, flexibility, and mental state, is there for emphasis only (or to look cool). Why the emphasis is there is generally lost to time. Who knows why the original maker of the form wants it there. People can guess, but without something in writing or without something passed on in oral history, nobody can know definitively the reason why.
To give you an example of why there's emphasis placed on a movement, it's sometimes done to mark the beginning of a throw. In these cases, this is the point in the form where you're emphasizing that the other person is now in trouble and about to be thrown. It's like saying, "Here we go!". You can see such an example in my breakdown of the Heian-Sandan form, here:
Name and meaning of stance where you stand with fists on hips?
In Heian-Sandan, you often pause a little after turning around and standing up in the middle of this form. The interpretation is that this is right after you've grabbed a hold of your opponent in preparation for an "o-goshi nage" throw. The pause is emphasizing that this is where the throw happens. It's a way of marking the end of something and the beginning of the next thing. In real life, you wouldn't actually pause there. You'd just complete the throw. And in the form, the pause generally doesn't need to be there. It's just that someone thought it should be there for emphasis. That's all.
Here's a video showing a demonstration of Heian-Sandan. The movement in question happens around 0:33...
https://www.youtube.com/watch?v=8OI-FTV3-4o&t=21s
My recommendation whenever someone asks why something is done slowly is to look back on old black-and-white movies or videos showing someone performing the form, and compare it against what the form looks like today. Also, compare different lineages and branches that all practice the same form and see how they differ. That can often lead you to discovering something. And of course, try to get good bunkai for it (look up "bunkai" if you don't know what it is).
Hope that helps.
When you fight, obviously you prefer to be fast over being slow. Everyone understands why you want to practice fast, but sometimes it's useful to practice slow. This may be different speeds within a form done the same way every time, or the same form done with variations in speed over different sessions.
You move slow when your technique does not yet support moving fast. For example, some styles dictate that the shoulders should not rotate independent of the hips, that your shoulders should remain aligned above your hips. If you have problems maintaining this alignment while moving fast, the solution is to practice (sometimes, at least) moving only as fast as you can while maintaining the alignment, which may be very slow.
Speed masks problems that become apparent when moving slowly. You can kick fast with poor balance and not fall because your kicking leg returns to the ground swiftly and catches you. But you cannot kick slowly with poor balance without stumbling. Moving slowly allows you to identify problems with balance, control, and continuity. It's also usually easier to correct these problems by practicing slowly initially.
I am not sure there should be a general way to answer the relaxation versus (dynamic) tension question. I think this should depend upon the particular element you are working on.
In the kung fu school where I studied, the instruction was explicit that after learning a form (sequence of movements independent of speed), you would vary the speed elements to suit your instantaneous state. The same form movement could be fast in one session, but slow in the next; there is no expectation that you will repeat the complete form exactly the same way. This training model is different from one where you can watch the form on YouTube and copy its performance to learn it.
As a metaphor, the form is a cup. You can have hot drinks or cold drinks in the cup. You can scald yourself with the first sip and decide you need to cool the drink down to finish it. You can decide to have your drink with sugar one day but without the next. You can start admiring the barista's cream pattern on top of your drink before deciding partway through that the now wonky pattern reminds you of something unpleasant and decide to stir the cream in. The cup is a vehicle for storing and consuming any beverage.
Maybe you start a form focusing on control and move very slowly. At some point, your body may be itching to go faster, so you let loose for a bit and blast through some movements. But then you notice that your weight is shifting when it shouldn't be, so you slow back down to focus on weight shifting. You think you have corrected the weight shifting, so you speed up to test your adjustments.
You can and should watch what other people drink from their cups to inform yourself how to make better drinks, but don't let the one time someone on YouTube drank a prune juice and kale smoothie out of their cup ruin your cup experience.
From my Kukkiwon 6th dan thesis (on poomsae and their benefits/analysis):
There are a few slow movements during the Taegeuk poomsae series, but there are a lot more in the black belt patterns. Their presence is undeniable, however the reason for having them does not seem to have a simple universal reasoning - unlike the other technical and philosophical areas of poomsae practice.
Most people have a simple explanation, there are slow movements purely for artistic reasons. The poomsae shown by Taekwondoin are not a violent pursuit and have a lot in common with a modern dance in terms of grace, balance and flowing movements. So some masters consider the slow movements in poomsae just that, an artistic expression in the middle of a pattern. Those same masters often advocate doing the movement slowly without tension, as a breathing exercise while developing a complete circle of hard/fast and soft/slow to show balance in the form.
A different set of masters see them as a physical pursuit, a chance to ensure every muscle is strong as the movement is performed with tension throughout the motion. By doing this contraction throughout the major muscle pairs (e.g. bicep and tricep are an antagonistic pair - each contracts moving the lower arm in the opposite directions) they are all given an equal workout, with the added benefit of maintaining the core tension, building strength in the minor stabilising muscles.
Finally, some grandmasters feel that the slow movements are a chance to build internal power. This is defined as the contraction and strengthening of the fascia throughout the body that forms connected lines from head to toe. It is believed by some that developing this internal power allows for strong movements without the use of “external power” (contraction of the major muscles). This is often practiced by Tai Chi or Kung Fu practitioners, but is not commonly taught in Taekwondo. Some Taekwondo grandmasters advocate the practicing of Taekwondo in a similar way to Tai Chi, slow and balanced rather than fast - practicing in this way may develop this fascia or connective tissue.
Personally I believe in a combination of the first two points of view. I believe that the slow movements add balance to the pattern and certainly aid in the artistic expression of Taekwondo. In addition to this, I believe they should be performed under tension to give the physical benefits to that style of contraction/movement. This allows grading examiners a good way to judge the confidence and skill of the candidate. Expert level practitioners should be confident in performing the movements to the required number of seconds, but lower level practitioners often either hurry them or use jerks in movement rather than smoothness to fill the time.
Various masters and grandmasters I interviewed during the writing of my theory. In personal conversations with me over the phone and instant messaging. One of them was my instructor Grandmaster Sim Pan, others didn't want to be named in my thesis (so doing it here would also go against their wishes). None of the opinions were stating it as fact, just their own opinions, so feel free to treat them with the weight of all internet opinions.
