I've read numerous sources indicating that improving core strength can improve your kicking power, especially for kicks that involve twisting. Has there been a rigorous exploration of the exact relationship between them? In particular, has anyone developed an exact mathematical model (presumably based on biophysics) of the relationship between muscle strength and kicking power?
Ok, so the concept to keep in mind is the bigger the muscle the more potential for power. Think of a horse and how those incredible strong legs can move so much more than that of a human. Then think of an elephant and how the power is enormous that they push down full grown big trees for fun thus maintaining the environmental atmosphere of the plains instead of letting them turn into forests.
Ok, with that in mind the largest muscles on the human body are the Rectus Abdominis muscle, External Abdominal Oblique muscle, Latissimus Dorsi muscle, and Gluteus Maximus core muscles. The issue is that these muscles are not direct striking muscles, but more protective and pivoting muscles which protect the torso structure and the organs underneath and the one that is the butt which allows us to sit. Where I'm sure we are all very thankful for this protection, the techniques and forms of martial arts are designed with body mechanics to harness these larger muscles and use them to channel additional strength and energy to kicks and punches.
Now all the abbendage muscles (i.e. arms/legs) are groups of corded muscles. if you watch body building this creates the ripples of muscles as you can see the individual muscles working together. That being said, no one muscle duplicates the function of another muscle. It's important to strengthen all related muscles to a specific movement as they each support their function. When looking at kicks specifically you can track the exact muscle usage and focus attention to those muscles specifically. Careful not to detract from other muscle sets those as when you change body position it changes muscles.
So, back to the biggest muscles and how to incorporate those. The position and muscle usage is key. If the position is off the muscles utilized are different. So the twisting of the hips and the movements are critical as those are the things that select the combination of muscles utilized for the body movement. That being said, all speed/strength comes from the muscles, so you are really just trying to select the right muscles. Thankfully systems of martial arts are developed that already do this positioning and you don't have to learn from scratch like the original practitioners did.
The "core" is focused on because that constitutes the main external muscles mentioned as well as the many supporting muscles that allow for muscle body and joint movement. The goal is a whip motion in that the force starts at the center or "core" and moves out through the supporting muscles to the point of impact. The more core power the more the final impact.
There is no "math" formula as there are so many varying factors depending on all the muscle conditions and usage it would be thousands of variations. The important thing is what is "your" formula. That is a body mechanics and personal training thing though and an instructor would need to help you work that out.
Fun thing to try is the belly thwomp or push. Naturally the more you have the easier it is, but this exercise uses those core muscles directly. Thus being able to push things with your belly is surprisingly strong even with the little movement that you do. Get a friend and try to push them with your belly, if you do it correctly with just the belly and not body checking them you will see that regardless of the person you are able to move them and displace their weight with your belly. That is the "core" power that is being harnessed in the kicks. It doesn't feel like alot, but the fact you can move them with such a small motion is powerful. The same power requires a full shove from an arm vs. a little nudge from the belly.
P.S. Please also note that injuries to muscles are often compensated for by varying the body movement again so other muscles more healthy can take over while the injured one heals. This compensating is sub-conscious and happens without you thinking about it, thus changing your formula to match.
P.P.S. Due to the amount of combination/variations if you want a specific set each kick needs to be done separately as there are way to many muscles and muscle combinations to indicate generically for "kicks".
NOTE: This is not about physical fitness. Strength and overall fitness of muscles should be addressed in the fitness SO as they are key points. Mass vs. Density and tone fits into the fitness category. This is merely talking about a muscle that is larger has more potential than a smaller muscle and the body mechanics to help incorporate larger human muscles into movements to generate more power. For a complete solution please consult muscle building and strength/endurance training for muscle development.
Kicking power is a dubious term. Some people who ask about power do not know what they mean when they use the phrase, and others do and want answers involving physics. Is force a result of multiplying mass times acceleration? Or is it a result of multiplying mass times velocity? Be careful here, you get two very different answers because of two very different applications.
I'll give a parallel example, in golf. The same applies to kicking, but it's easier to understand without the bias of subjective understanding of body mechanics and forces in martial arts.
Suppose Tiger Woods was at the tee, and the hole is 300 yards away. Does he use a putter or a driver? Both clubs have a vertical face, so what is the difference? When he swings a putter when on the green, his body moves with the putter as if the whole body assembly were sort of like a pendulum (that is, his center moves with the club head, and not ahead of it nor behind it). Would he swing the driver like a pendulum?
No, on both counts. Here's why:
The driver and putter do have vertical faces. But the driver is longer than the putter. That means, if his hands move at the same rate, the club head moves faster, because geometry tells us that two points about a radius, the one further from the center of the circle moves at a faster distance over time than the closer point as long as both points remain on the same radius. A golfer will tend to choose a lower numbered club head (longer shaft) than a shorter one and use the same swing when he needs a little more distance. He could swing faster with a shorter club and impart the same force, and not considering club face angles, the ball will land in the same spot - but that requires much more experience to pull off. It's easier to make the same swing and have a longer or shorter club (again, not considering club face angles).
Another thing about Tiger Woods: he would never drive a ball by swinging a driver like a pendulum as he would a putter. Why not? Isn't the longer driver going to drive the ball farther than the putter? Doesn't his core impart the same twisting force on the driver and putter? The answer is yes, the driver is longer and thus will have its head moving faster than the putter's head; and yes, the same twisting force (torque) is applied to both clubs when swinging as hard as he can. But that difference in head speed is negligible when the goal is to move the ball 300 yards. He needs more... power/speed/strength/force.
So how does Woods move the club head to get the ball to move 300 yards?
The answer is he needs more speed. He could go to Gold's Gym and work on those abs, but even the best body builders in the world don't have enough core strength to move a golf ball 300 yards by moving their body like a pendulum, no matter the size of a typical driver.
So, we can't increase the torque (or else body builders would be the best golfers, baseball hitters, and kickers in the world, agreed?)
But we can increase the speed of the club/leg by making use of potential energy. By twisting the body and pulling the club/leg, we are in essence whipping the thing around the radius at a much faster rate of speed then if we moved like a pendulum. Ever hear a golf or batting coach tell his protege to move the hips through the swing? He wants the hips to pass the ball before the arms do. This is just another way of saying twist the body: you can't twist the body without getting the hips to move through first.
And therein lies your answer: the core moves the body through the kick to help it gain speed (technically, it's velocity, not speed).
But we're not really done yet, because we don't know whether we are dealing with momentum or force. Or both.
Momentum is a result of multiplying the mass (the weight of the leg, although strictly speaking, it's not really the weight)... times the velocity (speed) of the moving leg: that's momentum.
Force is... well, that's complicated. Yes, it's mass times acceleration, I'm sure you hear that a million times. But does anyone really understand what that means? Momentum itself is all about the thing moving, but force is about the thing being moved. In other words, momentum is all about the club, and force is all about the ball.
To explain a little more, think of two kids playing a game of "toss the egg". One tosses an egg to the other, who must catch the egg without breaking it. If the catcher keeps his hand stiff and unmoving, and the thrower manages to get the egg into his hand, guess what's gonna happen to the egg? Yep: it'll crack. That's because the amount of time that egg hits the hand is a very short period of time. 100N has been applied to the egg if you catch it in 1/20 of a second to stop the egg's motion, and that'll crack it. But if you carried your hand in the direction of the egg's movement, and it takes a half second to catch it and stop it (instead of 1/20 of a second), only 10N has been applied, and might not crack the egg.
The catching hand, therefore, makes the difference in the amount of force needed to break the egg - not the tossed egg.
The more a kicker moves his kicking leg like a pendulum (not necessarily vertically where gravity plays a role - it doesn't - but where the body and the kicking leg are moving with the radius of the kick), the more core strength has relevance. The more a kicker relies on twisting force, the more core conditioning has relevance.
Bigger muscles don't necessarily translate to a faster kick - it can contribute to a slower kick. Developing fast-twitch muscles in the abdomen (conditioning), and making use of the joints like the knee (technique), and pulling the leg around to the target (technique) creates much more impulse force because it reduces the time of impact. This is like the egg-catcher who does not move the catch with the egg's motion.
A kick which moves slower imparts less impulse because of its increased contact time with the target. This is like the egg-catcher who moves with the egg's motion.
To directly answer your question ("is there an exploration of the relationship..."), yes, this is all embodied in Newton's laws of motion.
The method of improving core strength is the key: building muscle mass is less important than building the fast twitch reflexes to help change your technique, so that a whipping motion can be used. Building "core strength" necessarily requires defining the method of building "core strength".