ABSTRACT
As Chapter 5 highlights, the step-cycles of locomotion are not understandable in the terms of the conventional theory of auto-oscillatory phenomena for two major reasons. First, although the limbs are of fixed mass and length, the step-cycles can operate at indefinitely many near-stable periodicities. Second, this fact of indefinitely many near-stable periodicities implicates a flexible but lawful energy flow process for assembling and sustaining the locomotory step-cycles. With regard to the data of the main experiment, there are two outstanding questions whose resolution would appear to require a principled understanding of the energy flow processes responsible for the aforementioned facts of locomotion. Both questions have to do with the spacing of events: (1) What is the nature of the dependency in the steady state of a single left or right wrist-pendulum system's amplitude on its mass and length? and (2) what law governs the amplitudes assumed by a single left or right wrist-pendulum system of fixed dimensions when it is caused (by isochronous couplings) to assume different, stable periods? With respect to the first question, the data indicate that amplitude tends to vary inversely with mass and directly with length; they also suggest, however, that the signs of the exponents may be changeable from subject to subject. With respect to the second question, the data show that as the periodic time of a wrist-pendulum system 284 shortens, its amplitude sometimes increases, sometimes decreases, and sometimes remains constant within and across subjects. The present chapter and the next two ( Chapters 11 and 12 ) are devoted largely to a theory that accommodates these data. If the dynamics behind the spacing-timing linkage (question two) can be understood, then the link between amplitude and mass and length might become apparent (question one).
For a system's timing to vary, some physical parameter(s) must change. Systems whose timing varies as a function of variations in physical dimensions of the operational components are referred to as conditionally periodic systems. For the wrist-pendulum paradigm, the mechanical parameters of mass and length are prime candidates. When a wrist-pendulum system of fixed mass and length exhibits variations in timing, however, the "internal" candidate parameters are not so obvious. Of particular interest are the class of internal transformations defined as isothermal and adiabatic. These transformations require that the system's action and entropy production remain time-invariant throughout the transformation. The time-independence of action defines a constraint that scales the energy flowing into the system to the frequency of oscillation. The constraint operates as a soft conservational constraint. Ehrenfest's theorem defines the adiabatic linkage between a system's period of oscillation, energy carried within the oscillation and the mechanical parameter that is conditionally responsible for (a) bringing new energy into the system and (b) changing the system's operating period.
The chapter divides into two parts. The first part addresses conditional periodicity in open and closed conservative systems. The second part addresses the problem in the context of open nonconservative systems. Within each of these parts, equation sets are developed identifying adiabatic relationships between energy changes and frequency changes and between amplitude changes and frequency changes.
With respect to the energy by frequency relation, both conservative and nonconservative systems exhibit a positive first order linear relationship whose constant of proportionality identifies the action of the system. One notable feature of the nonconservative case is a nonzero, positive frequency intercept. Another is that the geometry of the nonconservative [energy X frequency] space exhibits symmetry with respect to the energy and frequency values in an adiabatic trajectory of states. The positive energy value coincident with twice the intercept frequency is equal to the value of the negative energy intercept. Moreover, whereas the conservative equation for action constancy links amplitude to period changes through a monotonic, nonlinear function, the resultant nonconservative equation links amplitude to period changes through a nonmonotonic, nonlinear function.
The physical concepts underlying these mathematical descriptions are discussed in detail. The focus is on understanding the physical basis for changes in coordinates of motion as a function of changes in mechanical parameters.
285The bottom line is that solutions to conditionally periodic problems do not necessarily require explicit formulation of the equations of motion if the transformations resulting in different periods satisfy adiabatic assumptions. Under these conditions the energy of oscillation varies strictly as a function of period and a constant of proportionality (an adiabatic invariant). Ehrenfest's theorem offers the most concise statement of the relationship between adiabatic invariants and mechanical transformability in conditionally periodic systems. To anticipate, the experimental evidence presented in Chapter 12 suggests that the coordinates of motion defining "comfort" states for wrist-pendulum motion are those that satisfy adiabatic assumptions. The goal of the next two chapters will be to unravel slowly the physical principles underlying adiabatic transformations as they bear on comfort states in conditionally periodic biomechanical organizations.
