Panther, Lion, and Dolphin
The butterfly swimming stroke is to most people the hardest stroke. It often comes down to a lack of the necessary strength to push harder.
When someone does the stroke in the lessons however, it isn't always the strength that I see they're missing. In their head they think if I push harder I'll get better. To a degree that is true, but it isn't the full picture.
Why + Philosophy:
Often for all strokes it comes down to timing of movements, and with things like breaststroke and butterfly timing is even more important. When you spend a lot of your time learning front crawl, and then move on to a harder stroke for a bit. Then you may not fully realize just how important timing is.
On top of that, the range of movement in butterfly is sometimes hard to achieve. As it takes flexibility and range, and not just strength.
How + Physics:
Now that I have gotten the preconceived notions out of the way. How do you do it?
You mimic these three animals:
Panther
Lion
Dolphin
Starting out in the gliding/streamline with your hands out above your head together, they need to get yourself moving. Since they are already ready to pull the water, then you don't need to do anything else. Pulling is all you need.
Pull towards your stomach like a panther runs with their legs.
From there you need to reset, and get back to the top for next pull. Sliding up along the sides of your body. Your arms reach a 90 degree angle, as they come out of the water HIGH. One could say it is like a double front crawl, as both arms are coming out at the same time.
Your arms are up like a lion about to pounce on it's prey.
Finally you have the descent. You need to come from the air to the water, and just like the front crawl you don't want to SMACK. So diving your hands together into the water, which gets you back into a streamline position. In which case you can dolphin.
You'll be doing the dolphin kick the entire time, but now your hands have essentially become the snout.
What + Psychology:
This example of how the movement works helps you imagine how your body is supposed to look when doing the strokes.
By keeping in the 3 step kind of system, our brains like the chunks making it easy to remember like the 3 simple steps to swimming from before.
Conclusion
Remember run like a panther, pounce like a lion, and dive like a dolphin!