# Fractional Dynamics

It may sound like a misconception referring to a dynamics-related metric in a section addressing kinematics. However, it is not. This section has presented a set of computational metrics revolving around the kinematics of the athlete, i.e., metrics that were computed based on the motion of the athlete or its body segments, without considering the forces that cause them to move. This section is no different since the concept of fractional dynamics implies the use of fractional calculus to describe the trajectory of football players, therefore ignoring any forces leading inherent to it.

Only a few numbers of applications based on fractional calculus have been reported so far within sport sciences literature. One of them was the development of a correction metric for golf putting to prevent the inaccurate performance of golfers when facing the golf lip out phenomenon (Couceiro, Dias, Martins, & Luz, 2012). The authors extended a performance metric using the Griinwald- Letnikov approximate discrete equation to integrate a memory of the ball’s trajectory. The same authors further applied the concept of fractional calculus in football, be it to improve the accuracy of tracking methods by estimating the position of players based on their trajectory so far (Couceiro, Clemente, & Martins, 2013) and to characterize the predictability and stability levels of players during an official football match (Couceiro, Clemente, Martins, & Machado, 2014).

The previous sections supported the idea that football is a complex dynamic system, wherein the motion of each player is usually chaotic and difficult to predict (Grehaigne, Bouthier, & David, 1997). Under those assumptions and taking into account that the fractional derivative can be considered as a natural extension, or generalization, of the integer (i.e., classical) derivative, it presents itself as an excellent instrument for the description of ecological memory and hereditary properties of processes (i.e., path dependency).

To start with, let us consider the discrete case in which the motion of player

*i *may be defined as follows:

in such a way that the difference between the position of player *i* at time f+1 and time f, for a period of 1 second, is equal to its current velocity vector *Vj[t+*1]. Hence, *x,* [f +1] — x, [/] corresponds to the first-order integer difference. By adopting the approximate discrete-time Griinwald-Letnikov fractional difference of order *a,* [f], generalized to a real number 0 < *a*, [(] < 1, and for a sampling period of 1 second and a truncation order r, the position of player *i* at time t + 1 can be written as (Couceiro, Clemente, & Martins, 2013) follows:

with Г being the gamma function and *x,* [f +1] being the approximated position of player i at time f + 1. It should be noted that such strategy increases the memory complexity as it requires memorizing the last *r* positions of each player, i.e., О [/N,5]. Nonetheless, the truncation order r does not need to be too large and will always be inferior to the current time *t,* i.e., r < *t.*

As one may observe in Equation (4.12), a problem arises regarding the calculation of the fractional coefficient Ct, [f]. A player’s trajectory can only be correctly defined by adjusting the fractional coefficient *a*, [f] along time. In other words, Of, [f] will vary from player to player and over time. Hence, one should find out the best fitting O', [f] for player *i* at time f based on its last known positions so far. The value of O', [f] will be the one that yields a smaller error between the approximated position *Xj* [f +1] and the real one x, [f+l], denoted as *d' ^{mn}.* This reasoning may be formulated by the following minimization problem:

The solution of Equation (4.13) can be found with any optimization method, such as golden section search and parabolic interpolation (Brent, 1973; Forsythe, Malcolm, & Moler, 1977). By analysing the effect of *a,* [f] in Equation (4.2), one can conclude that the closer to 1 the values of [f] are, the higher predictable player *i* is. In other words, a value of Of, [f] = 1 means that Equation (4.12) can accurately predict the next position based on the previous ones, i.e., *x]* [f +1] *=Xj* (7+1]*d' ^{nm}* (ct, [f]) = 0. Therefore, for constant and linear trajectories, i.e., without moving at all or with constant speed, the fractional coefficient a, [f] gets closer to a constant value of 1 - being highly predictable. For a chaotic trajectory, the fractional coefficient variability decreases considerably, presenting values close to Of, [f ] = 0.4 in some situations. This variability is only exceeded by the random trajectories, in which the fractional coefficient in some situations may even get close to O', [f] =0 (Couceiro, Clemente, Martins, & Machado, 2014).