On the other hand...

In Taekwon-Do, the use of the reaction hand is almost universal. It is used as a counterweight during the execution of most strikes and blocks. And it based on one of the simplest and most universal physical laws we have; Newtons third law of motion:

When a body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction to that of the first body.

In the case of the reaction hand, the relevant interaction is between the reaction hand/arm (body A), and the centre of mass (body B), which is located at the centre of the body, slightly below the navel.

From the third law we see that:

1. When the hand is pushed outwards from the centre of mass, the centre of mass is also pushed away from the hand.¹
2. When the hand is pulled towards the centre of mass, the centre of mass is also pulled towards the hand.
3. When the hand accelerates laterally with respect to the centre of mass, the centre of mass gains momentum in the opposite direction. This is also true at any instant where the hand is orbiting the centre of mass.

The Encyclopaedia of Taekwon-Do:

"Another reaction force is your own. A punch with the right fist is aided by pulling back the left fist to the hip."

This seems pretty straight-forward, between the three applications of the third law I have stated here, and the reaction force quote form the encyclopaedia, it should be easy to understand the use of the reaction hand.

But a little research shows that this is widely misunderstood, and often written off as trivial.
The reason for this is simple; if you don't begin by understanding the laws of motion you wont know when you are violating them. The vast majority of us start form the above quote where the punch is 'aided', or similar instruction from a teacher, which, despite being correct, tells us exactly nothing about how the left fist aids the right punch.
And on that vague foundation we make assumptions, and those assumptions may, and do, lead to misunderstandings.

The following are each examples of either misunderstanding, misapplication, or both:

1.

From C'hang Hon Taekwon-Do Hae Sul by Mr. Stuart Anslow:

"However, when there is an actual opponent, the equal and opposite reaction takes place within the opponent themselves, so where as in training, the reaction hand has a purpose, in combat this purpose is voided..."

Mr. Anslow's understanding appears to be that the purpose of the reaction hand is to stop or slow the technique, just as hitting a target would stop or slow the technique.

This is one of the errors that contribute to the contact-time fallacy; the idea that  stopping a punch yourself (pulling a punch) is the same thing as having it stopped for you (hitting a target).

Mr.Anslow is wrong here, but his mistake is probably based on an explanation given by General Choi himself in his encyclopaedia, which also incorrectly reverses the role played by the reaction hand.

The encyclopaedia is not the only book that has the physics right on one page and backwards on another, and before I look at Choi's mistake I'll take a look at a very similar situation in a much more recent publication.

 2.

This time its First Grandmaster Rhee's 'This Is Taekwon-Do Vol.1' in the firing line.

In fact, This is Taekwon-Do Vol.1 has an excellent introductory section on the reaction force, and has a couple of examples that very clearly demonstrate how it works.
However, it also has a classic mistake in the very first example given, the example of the front punch;

Reaction force diagram from This Is Taekwon-Do by Grandmaster Rhee.Ki Ha

"One arm pulls whilst the other pushes around the pivoting points which are the spine and hips, you are using opposing sides of your body to generate power. If you simply punched without the reaction hand moving, you would only be using power from one side of your body."

This explanation, with the help of the top-down diagram, states that the action arm and reaction arm work together to rotate the body, and this rotation increases the power of the punch.

But this is clearly pulling yourself up by the bootstraps; the action of the arms rotates the body, the body rotation aids the action of the arms, which rotates the body, which aids the arms..... it's an infinite loop.

If we look at this model with the third law of motion in mind we can see that the action arm and reaction arm both work together to turn the body in the opposite direction:

• As the punch goes forward it pushes the action side of the body backward.
• As the reaction hand is pulled backwards it pulls the reaction side of the body forward.  

This would rob the punch of power, instead of aiding it.

This is not a trivial point. I think it is exactly this broken model, specifically treating the front punch as something that develops its power through rotation about a central axis, that has led many martial artists from Taekwon-Do and Karate to abandon the use or understanding of reaction force.

This is the first learned and most obvious use of the reaction hand, and if it is doing exactly the opposite of what it is supposed to be doing, then it is no surprise that it never really feels right to the practitioner.

A better model tells a different story.
The action performed when executing a front punch is more akin to a closing door, with the reaction hip remaining largely stationary as the opposite 'action' hip comes forward.
The centre of mass lies between these two points and so travels forward with the action hip. This is how we get body mass into the strike, not by rotating around the centre of mass, but by moving the centre of mass in the direction of the strike.

The purpose of the reaction arm is to pull the body forward, even as the punching arm attempts to push it back. This is the counterbalance; the striking arm robs us of forward momentum, the reaction hand adds forward momentum.

This application of reaction force is so deeply embedded in Taekwon-Do that nearly every formal technique has been designed around it.
The trajectory of the reaction hand usually attempts to mirror the trajectory of the action hand,  so in Won Hyo, for example, we see the reaction hand pulled across the chest to the opposite shoulder rather than to the hip for inward knife-hand strike in the L-stance.

In the encyclopaedia Choi adds an additional reason to bring the hand to the opposite shoulder instead of the hip:

3.

From The Encyclopaedia of Taekwondo:

"Incorrect: The attacking tool is over extended, passing the point of focus, because the other fist is pulled to the hip instead of the shoulder."

It would be easy to interpret this to mean that the purpose of the reaction hand, when pulled to the shoulder, is to stop the strike at the point of focus, and this is probably where Mr. Anslow picked up the idea.
But that's not a valid interpretation, simply because that is not the purpose of the reaction hand. Instead, Choi is giving an additional reason why the to-the-hip reaction is less suitable for this strike.

The claim Choi is making here is that pulling the fist to the hip will cause that hip to move backwards, resulting in an unwanted rotation of the torso.
Unfortunately this is the same faulty physics found in the earlier front punch example. No amount of reaction force will push that hip back, because the reaction force is operating in the opposite direction; it is pulling the hip forward.

In fact, pulling to the hip works well here, but pulling to the shoulder works better as it more closely mirrors the lateral trajectory of the strike at the moment of impact.

Hiki-te.

Using the reaction hand as a counterweight, as something to pull against, ties in perfectly with a concept borrowed from Karate; hiki-te, the pulling hand.

In many, though not all, techniques the reaction hand can be used to grab and pull the opponent into a strike. The mechanic is the same as the standard reaction hand, but now the opponent provides much better anchorage for pulling into the strike, while the pulling action also unbalances and/or positions the opponent.

I won't argue that using the non-striking hand is always best utilised as a counterweight, pushing, pulling, holding and guarding are all options, though a good guard is not near as important in self defence as it is in combat sport. But that's a point for another post.

My favourite example of  reaction force use is Lee Morrison in action. Partially because he does it so well, but also because Lee really does not like traditional martial arts (making him an impartial judge of traditional technique, if not a downright hostile one) and is interested only in what works for him.

Take a look here, specifically watching the reaction hand and how he pulls it both to the side (not to the hip, but that's a matter of degree) and to the opposite shoulder/chest, at different times:

Footnotes.

1. I'm treating the hand as a kind of point mass representing the mass of the whole arm, which is probably the original convention, hence 'reaction hand ' instead of 'reaction arm'.