Investigating a Nonconservative Invariant of Motion in Coordinated Rhythmic Movements

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Abstract
The dimensions of an animal's limbs are fixed, but in locomotion and other rhythmic activities, they oscillated at a number of different frequencies. How might the physical conditions for this frequency variation be characterized? Kugler and Turvey (1987) hypothesized that the conditions might be adiabatic. A rhythmic system undergoes an adiabatic transformation of frequency when the stiffness is changed without a transfer of energy by heating. By standard definitions, adiabatic transformability is achievable only in conservative systems and only at infinitely slow rates of transformation. Kugler and Turvey's (1987) hypothesis extends adiabatic to systems that are dissipative and transformed rapidly by internal sources of energy, such as biological movement systems. Two predictions follow from the hypothesis. The first prediction is that a relation should be obtained in frequency-energy coordinates that has constant slope (Ehrenfest's adiabatic relation, a semipermanent invariant of motion) and an energy intercept less than zero (constant energy dissipation regardless of frequency). The second prediction is that the positive linear relation in frequency-energy coordinates can be satisfied by different relations in period-amplitude coordinates; amplitude increasing, amplitude increasing then decreasing, amplitude decreasing. The predictions were evaluated in four experiments with the same three participants. In each experiment, the rhythmic movement unit was defined by a pendulum of fixed dimensions held in the right hand that was made to oscillate at frequencies in the range 0.6 Hz to 1.8 Hz by the requirement of 1:1 frequency locking with a pendulum of different dimensions held in the other hand. Changes in the period of a pendular rhythmic movement were accompanied by statistically significant changes in the amplitude. Amplitude's systematic dependence on period differed, however, among the three participants in the experiments.