# Tai chi and calculus

Tai chi as a martial art seems far removed from actual fighting, since the motions are slow to the point of static.  However, a few years ago during a conversation with Debra Page we came up with a cool analogy to describe part of why studying tai chi is valuable to the full speed case.  I never got around to writing it up, so here goes: tai chi is like calculus.

The fundamental idea of calculus is that change is simpler when viewed in small amounts.  A smooth function, evaluated at a variety of points spaced a finite distance apart, can contain all manner of nonlinearities, jumps, wiggles, etc.  However, if you look close enough, every smooth function turns into the simplest possible case: a straight line.  The beauty is that knowledge gained by looking closely around each point can be extrapolated from infinitesimal up to finite to understand the global behavior of the function.

Now from calculus to physics: if we understand the behavior of a dynamical system for small intervals of time and space, calculus lets us jump from infinitesimal to finite to understand the behavior of the system for larger motions.  In fact, we can go one step further: d’Alembert’s principle tells us that in order to understand the infinitesimal behavior of a dynamical system, it suffices to understand the infinitesimal behavior of an equivalent static system.  To do this, we start with a snapshot of the position of a system at a given time.  In the general case our snapshot won’t be static since the forces won’t sum to zero, but we can compensate for this by subtracting so called “inertial forces” from our system to balance the rest of the forces.  Once everything in balance, and our snapshot has been reduced to a static system, we can read off the dynamical behavior immediately: the inertial forces turn out to be exactly mass times acceleration.

Now say we’re in a fast martial art, such as karate, and we’d like the understand the process of stepping from one stance to another.  We start with weight on two feet, and in one smooth motion remove the weight from one, move it through some chosen path to a new position, and redistribute weight into the new stance.  There are innumerable aspects we’d like to understand about this process: the energy required, its stability under unexpected external stimuli, the effort required to change direction if suddenly necessary, the amount of force that can be exerted in any particular direction at any point along the motion, etc.  Calculus teaches us that to gain this understanding, it suffices to study the behavior of the system over small periods in time and space.  Unfortunately, a human is a fairly high dimensional system, and the linear functions we have to understand are high dimensional as well.  It’s very difficult to analyze a high dimensional space while moving quickly between stances.

D’Alembert to the rescue: instead of applying calculus to the dynamical system, we move from one stance to another extremely slowly, taking time to analyze the linear space around each nearly static pose.  Additional muscle action may be required, corresponding to the inertial forces necessary to turn the dynamic case static.  Now that we have time, many of the questions we’d like to understand about the fast motion translate into experiments we’re almost forced to run: e.g., compensating for any wiggles from our path corresponds to running a stability analysis.  Want to know if it’s possible to change direction on a dime halfway through the motion?  Just stop in the middle, move a little bit in the given direction, and feel the floor to see if the required force is normal or tangential.  With enough repetition, and enough slowness, we’ll inevitably sample enough of each linear space to get an intuitive feel for the detailed structure of the motion, knowledge that can be applied directly if we ever need to scale back up to fast.

Caveat: my total hours of tai chi training can be counted on two hands, but Debra knows far more and liked this analogy, so hopefully this post is reasonable.