In Heinrich Hertz's account of conservative systems, the concept of a cyclic coordinate, and, in particular, what he called an adiabatic cyclical system, enters as an important technical tool. This chapter explains the meaning of these concepts as understood in the ordinary image of mechanics. First, the historical development of the concept and its important mechanical and mathematical properties are discussed. An example of a simple mechanical system that has a cyclic coordinate is then presented. This example is only meant as a didactical device. The chapter concludes by examining Hermann von Helmholtz's use of cyclic motion in his mechanical model of thermodynamics. Helmholtz's papers on this topic are important for the present book because they were the point of departure for Hertz's ideas on cyclic coordinates. This chapter also considers Edward John Routh's modified Lagrangians and J.J. Thomson's equation of hidden cyclic motion.
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