This question of engaging the hips made me think of the term often usen in Japanese: "koshi wo ireru" - to "put in" the hip/hips. We sometimes use the term "put your back into it" when we want to emphasize the concept of putting some oomph into a physical effort. For the sake of experimenting, I googled the term in Japanese - in romaji. The following text popped up. It is from a group that dance Nihon Buyo. A fascinating read, by the way.
"Is the use of the lower back and hips one of the elements in the aesthetic beauty of Nihon Buyo?
Well, we speak of "applying the lower back and hips" (koshi wo ireru) in Nihon Buyo, but I don't think even the greatest master can explain what that means in a word. It isn't simply a matter of the lower back and hips (koshi) coming into play when you take a bent-knee stance, but you might say it gives you more stability, or that it plants you more firmly on the ground. An example of applying the lower back and hips (koshi wo ireru) can be seen in the familiar scene in Sumo wrestling when one wrestler tries to lift another but he can't be moved. It is if the wrestler has suddenly become heavier and can't be lifted. That wrestler can't be picked up because he is in a state of applying the lower back and hips. Another example is when you try to carry a cup that is filled to the brim with water without spilling it. We instinctively lower our center of gravity as we walk with the cup, don't we? That is also a state of applying the lower back and hips. It is hard to explain in words but it is a sense we Japanese as an agrarian people have acquired naturally."
I just wanted to share this with you.