The notion of relative phases plays an important role in synergetics approaches to movement science. They act as order parameters in the sense that a stable relative phase between the movements of several limbs characterizes a stable coordinative pattern (Kelso, 1995). In the theory of game sports a mayor problem is to find a description for their most essential property, the interaction process between the two players. A dynamical system view adopts the notion of relative phases and uses it to describe the spatial interactions of the two opponents in the court (Mc Garry, et al., 2002; Palut & Zanone, in press). In this study the potential of relative phase to detect critical situations in tennis rallies was examined. Methods In a first approach a video-taped rally of a match between Justine Henin and Serena Williams at the French Open 2003 was analysed. It lasted 18 seconds and showed a typical behaviour in base-line tennis. The lateral distance from the centre - line of the court was taken every 0,2 seconds for the two players. Relative phase was calculated according to a method described by Pikovsky et al. (2001): The movements of the players were described in a phase plane (position vs. speed, see fig. 1). Cycles were defined from stroke to stroke. The trajectory in the phase plane was transformed on a unit cycle. Relative phase was calculated as the difference between the momentary angular position of the two players in the unit cycle. The results show that relative phase is rather stable during stable periods of the rally (see fig. 2). Two critical situations were indicated by dramatic changes of relative phase: With her third stroke Serena started a cross play forcing Justine to the backhand side, and her fifth stroke was a backhand longline creating the decisive pressure. Discussion In this first attempt to study critical situations in tennis by means of relative phases, it has turned out to be a promising tool. The spatial interactions of the two players incorporate very much what is essential to the game. Many problems remain to be solved, however. There are for example several alternative methods to compute relative phases. The data used to describe spatial interaction should comprehend also a second dimension, the approach to the net. Finally it would be useful to have fast procedures for data collection. Studying tennis as a dynamic system using relative phases is not only theoretically intriguing, but promises also to provide a methodological approach to previously hardly measurable, but essential aspects in tennis and other game sports: creating pressure, dominating a rally, recover from pressure, and preparing the winning stroke. Thus, it may be expected that this new approach will be of great practical impact.
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